Synthetic control with pymc models#

import arviz as az

import causalpy as cp
%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = 'retina'
seed = 42

Load data#

df = cp.load_data("sc")
treatment_time = 70

Considerations#

Donor pool selection#

Before fitting, it is good practice to inspect pairwise correlations among units in the pre-treatment period. Negatively correlated donors should be excluded to avoid interpolation bias [Abadie et al., 2010, Abadie, 2021].

pre = df.loc[:treatment_time]
corr, ax = cp.plot_correlations(pre, columns=["a", "b", "c", "d", "e", "f", "g"])
ax.set(title="Pre-treatment correlations")
[Text(0.5, 1.0, 'Pre-treatment correlations')]
../_images/42e5b6e9430accbfe9e59aa39f9ed10cb476dc1b6ba74d5d63e233f742e6dbf0.png

All control units are positively correlated with the treated unit (actual), so no donor pool curation is needed for this dataset. In practice, you would exclude any controls with negative or near-zero correlations.

Convex Hull Assumption#

The synthetic control method uses non-negative weights that sum to one. This means the synthetic control is a convex combination of the control units—it can only produce values within the range spanned by the controls at each time point.

Note

For the method to work well, the treated unit’s pre-intervention trajectory should lie within the “envelope” of the control units. If all controls are consistently above or below the treated unit, the method cannot construct an accurate counterfactual.

CausalPy automatically checks this assumption when you create a SyntheticControl object and will issue a warning if violated. See the Convex hull condition glossary entry and Abadie et al. [2010] for more details. The Augmented Synthetic Control Method [Ben-Michael et al., 2021] can handle cases where this assumption is violated.

Observation noise prior#

The default WeightedSumFitter places a \(\mathrm{HalfNormal}(1)\) prior on the observation noise \(\sigma\). This prior assumes the data is approximately on unit scale. If your outcome variable has values much larger than 1 (e.g., daily revenue in the thousands), the prior will likely severely constrain \(\sigma\), producing confidence intervals that are much too narrow.

By default, SyntheticControl automatically rescales this prior to match the scale of the treated-unit data (auto_scale_sigma=True). If you need the original HalfNormal(1) default, pass auto_scale_sigma=False. You can also specify a fully custom prior via the WeightedSumFitter priors parameter.

Run the analysis#

Note

The random_seed keyword argument for the PyMC sampler is not necessary. We use it here so that the results are reproducible.

df.head()
a b c d e f g counterfactual causal effect actual
0 0.806091 -0.271100 0.664254 1.705922 -1.449230 0.785439 -1.234083 -0.232966 -0.0 -0.076404
1 1.442677 0.462594 0.559940 1.862041 -1.615366 0.386238 -1.798470 -0.069413 -0.0 -0.063946
2 1.916486 0.845719 0.894046 2.061073 -1.978696 0.971803 -0.490287 0.095440 -0.0 0.392813
3 2.001511 1.067162 0.288606 1.934534 -2.093951 1.889334 -0.478545 0.261716 -0.0 0.133106
4 2.662849 1.243252 0.966345 2.482698 -1.407598 2.184504 -0.434592 0.429432 -0.0 0.293294
result = cp.SyntheticControl(
    df,
    treatment_time,
    control_units=["a", "b", "c", "d", "e", "f", "g"],
    treated_units=["actual"],
    model=cp.pymc_models.WeightedSumFitter(
        sample_kwargs={"target_accept": 0.95, "random_seed": seed}
    ),
)
Hide code cell output
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [beta, y_hat_sigma]
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  ━╺━━━━━   376    0         0.017     255        148.41    0:00:02   0:00:11   
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  ━╺━━━━━   344    0         0.017     511        135.87    0:00:02   0:00:13   
                                                  draws/s                       
  ━╺━━━━━   345    0         0.033     127        136.22    0:00:02   0:00:13   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   327    0         0.020     255        124.44    0:00:02   0:00:14   
                                                  draws/s                       
  ━╺━━━━━   389    0         0.021     31         147.49    0:00:02   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   353    0         0.019     127        133.80    0:00:02   0:00:13   
                                                  draws/s                       
  ━╺━━━━━   360    0         0.025     191        136.54    0:00:02   0:00:13   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   339    0         0.017     511        123.61    0:00:02   0:00:14   
                                                  draws/s                       
  ━╺━━━━━   401    0         0.021     255        145.93    0:00:02   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   365    0         0.022     255        132.69    0:00:02   0:00:13   
                                                  draws/s                       
  ━╺━━━━━   378    0         0.023     127        137.53    0:00:02   0:00:12   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   355    0         0.023     127        123.70    0:00:02   0:00:14   
                                                  draws/s                       
  ━╺━━━━━   415    0         0.027     63         144.80    0:00:02   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   384    0         0.027     127        134.13    0:00:02   0:00:13   
                                                  draws/s                       
  ━╺━━━━━   397    0         0.024     127        138.98    0:00:02   0:00:12   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   371    0         0.025     255        124.45    0:00:02   0:00:14   
                                                  draws/s                       
  ━╸━━━━━   430    0         0.029     63         144.35    0:00:02   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   398    0         0.017     255        133.84    0:00:02   0:00:12   
                                                  draws/s                       
  ━╺━━━━━   417    0         0.029     63         140.15    0:00:02   0:00:12   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   388    0         0.022     127        125.53    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   446    0         0.025     255        144.47    0:00:03   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   411    0         0.023     127        133.14    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   435    0         0.022     255        140.97    0:00:03   0:00:12   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   401    0         0.018     191        125.48    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   462    0         0.024     255        144.61    0:00:03   0:00:11   
                                                  draws/s                       
  ━╺━━━━━   427    0         0.018     255        133.64    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   455    0         0.021     63         142.54    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   414    0         0.016     255        125.23    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   477    0         0.017     63         144.14    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   441    0         0.016     127        133.48    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   471    0         0.021     383        142.68    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╺━━━━━   426    0         0.023     127        124.69    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   491    0         0.022     15         143.75    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   454    0         0.021     511        132.80    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   490    0         0.030     127        143.38    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   442    0         0.020     255        125.45    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   503    0         0.017     63         142.56    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   466    0         0.021     127        132.07    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   508    0         0.023     63         144.39    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   453    0         0.018     127        124.53    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   517    0         0.014     63         142.09    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   479    0         0.020     31         131.82    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   518    0         0.021     31         142.78    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   472    0         0.017     63         125.91    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   530    0         0.017     255        141.37    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   491    0         0.021     159        131.10    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   531    0         0.023     63         142.11    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   485    0         0.017     119        125.56    0:00:03   0:00:13   
                                                  draws/s                       
  ━╸━━━━━   544    0         0.019     191        140.86    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   505    0         0.026     255        130.94    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   542    0         0.018     31         141.04    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   500    0         0.019     127        126.18    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   563    0         0.016     127        141.76    0:00:03   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   515    0         0.019     127        129.85    0:00:03   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   553    0         0.017     255        139.52    0:00:03   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   514    0         0.026     63         125.85    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   576    0         0.020     71         141.20    0:00:04   0:00:11   
                                                  draws/s                       
  ━╸━━━━━   528    0         0.021     127        129.41    0:00:04   0:00:12   
                                                  draws/s                       
  ━╸━━━━━   567    0         0.015     127        139.27    0:00:04   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   523    0         0.024     319        124.89    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   593    0         0.020     127        141.51    0:00:04   0:00:10   
                                                  draws/s                       
  ━╸━━━━━   539    0         0.024     191        128.89    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   581    0         0.016     127        138.68    0:00:04   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   538    0         0.032     159        125.08    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   609    0         0.022     31         141.35    0:00:04   0:00:10   
                                                  draws/s                       
  ━╸━━━━━   552    0         0.019     159        128.26    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   597    0         0.017     255        138.76    0:00:04   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   557    0         0.024     31         126.40    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   626    0         0.025     127        141.79    0:00:04   0:00:10   
                                                  draws/s                       
  ━╸━━━━━   566    0         0.021     127        128.34    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   607    0         0.015     255        137.66    0:00:04   0:00:11   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━╸━━━━━   571    0         0.025     127        126.19    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   639    0         0.025     191        141.21    0:00:04   0:00:10   
                                                  draws/s                       
  ━━╺━━━━   582    0         0.024     127        128.74    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   624    0         0.013     31         138.06    0:00:04   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   585    0         0.024     127        126.10    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   655    0         0.026     31         141.14    0:00:04   0:00:10   
                                                  draws/s                       
  ━━╺━━━━   593    0         0.024     319        128.05    0:00:04   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   634    0         0.017     255        136.97    0:00:04   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   598    0         0.027     127        125.68    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   670    0         0.030     63         140.87    0:00:04   0:00:10   
                                                  draws/s                       
  ━━╺━━━━   608    0         0.022     63         128.00    0:00:04   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   648    0         0.016     255        136.57    0:00:04   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   610    0         0.023     127        125.40    0:00:04   0:00:12   
                                                  draws/s                       
  ━━╺━━━━   688    0         0.021     63         141.53    0:00:04   0:00:10   
                                                  draws/s                       
  ━━╺━━━━   623    0         0.017     83         128.27    0:00:04   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   663    0         0.020     95         136.66    0:00:04   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   623    0         0.016     511        125.50    0:00:04   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   704    0         0.020     255        141.63    0:00:04   0:00:10   
                                                  draws/s                       
  ━━╺━━━━   636    0         0.018     159        127.96    0:00:04   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   673    0         0.019     127        135.73    0:00:04   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   636    0         0.022     255        125.05    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   722    0         0.022     255        142.32    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╺━━━━   647    0         0.019     151        127.31    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   688    0         0.017     127        135.55    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   652    0         0.024     31         125.41    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   735    0         0.021     255        141.50    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╺━━━━   664    0         0.021     127        127.78    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   699    0         0.020     127        134.60    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   666    0         0.025     255        125.34    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   749    0         0.019     255        141.07    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╺━━━━   679    0         0.023     127        128.03    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╺━━━━   711    0         0.019     255        134.01    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   683    0         0.025     63         125.99    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   758    0         0.017     23         140.20    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╺━━━━   699    0         0.020     127        128.94    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   723    0         0.016     255        133.37    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   697    0         0.020     255        125.90    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   770    0         0.014     255        139.15    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╺━━━━   709    0         0.020     255        128.17    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   735    0         0.017     255        132.93    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╺━━━━   706    0         0.022     127        125.19    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   779    0         0.018     255        138.09    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   720    0         0.017     255        127.66    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   747    0         0.017     255        132.46    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   721    0         0.024     127        125.09    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   794    0         0.015     127        137.93    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   734    0         0.021     255        127.63    0:00:05   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   760    0         0.016     47         132.18    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   736    0         0.020     175        125.40    0:00:05   0:00:11   
                                                  draws/s                       
  ━━╸━━━━   803    0         0.017     511        136.99    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   749    0         0.016     255        127.63    0:00:05   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   769    0         0.016     255        131.01    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   754    0         0.021     79         125.97    0:00:05   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   810    0         0.019     255        135.48    0:00:05   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   761    0         0.020     63         127.31    0:00:05   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   778    0         0.021     95         130.35    0:00:05   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   769    0         0.023     127        126.29    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   824    0         0.023     127        135.40    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   776    0         0.021     127        127.42    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   787    0         0.022     191        129.26    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   785    0         0.019     31         126.48    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   839    0         0.024     255        135.28    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   786    0         0.014     255        126.71    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   799    0         0.024     127        129.05    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   800    0         0.023     63         126.60    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   855    0         0.027     127        135.34    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   794    0         0.017     511        125.69    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   814    0         0.027     127        128.95    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   813    0         0.024     127        126.41    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   869    0         0.027     255        135.26    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   806    0         0.018     63         125.41    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   826    0         0.025     63         128.56    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   827    0         0.016     255        126.69    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   881    0         0.021     63         134.94    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   815    0         0.021     255        124.86    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   840    0         0.018     255        128.55    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   837    0         0.017     127        125.88    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   896    0         0.021     255        134.76    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   826    0         0.022     143        124.33    0:00:06   0:00:10   
                                                  draws/s                       
  ━━╸━━━━   848    0         0.020     255        127.64    0:00:06   0:00:10   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━╸━━━━   848    0         0.017     167        125.70    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   907    0         0.024     223        134.20    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   840    0         0.021     127        124.39    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   862    0         0.019     255        127.73    0:00:06   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   858    0         0.019     83         124.84    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   914    0         0.022     31         133.11    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   848    0         0.020     255        123.45    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   870    0         0.013     127        126.72    0:00:06   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   871    0         0.016     255        124.72    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   926    0         0.024     127        132.67    0:00:06   0:00:09   
                                                  draws/s                       
  ━━╸━━━━   857    0         0.023     255        122.71    0:00:06   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   878    0         0.015     127        125.75    0:00:06   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   885    0         0.017     127        124.68    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   939    0         0.026     255        132.32    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   874    0         0.021     255        123.33    0:00:07   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   884    0         0.015     383        124.68    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   898    0         0.019     63         124.71    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   957    0         0.024     127        132.80    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   889    0         0.021     127        123.36    0:00:07   0:00:10   
                                                  draws/s                       
  ━━━╺━━━   893    0         0.019     319        124.07    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   909    0         0.018     255        124.21    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   970    0         0.025     127        132.55    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   902    0         0.025     255        123.29    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   903    0         0.019     255        123.54    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   919    0         0.015     63         123.76    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   984    0         0.023     127        132.49    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   917    0         0.028     31         123.53    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   914    0         0.019     63         123.19    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   935    0         0.019     255        123.90    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   993    0         0.025     255        131.76    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   937    0         0.026     127        124.23    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   931    0         0.023     127        123.60    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   946    0         0.020     255        123.66    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1004   0         0.027     255        131.31    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   955    0         0.028     63         124.91    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   943    0         0.020     255        123.36    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   960    0         0.021     191        123.75    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1015   0         0.027     255        130.89    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   969    0         0.026     63         125.12    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   955    0         0.024     255        123.29    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   973    0         0.020     127        123.73    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1027   0         0.027     191        130.68    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   989    0         0.029     31         125.84    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╺━━━   968    0         0.020     255        123.22    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   984    0         0.014     271        123.47    0:00:07   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1041   0         0.027     15         130.65    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1002   0         0.024     511        125.90    0:00:07   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   977    0         0.024     63         122.66    0:00:07   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╺━━━   995    0         0.018     511        123.09    0:00:08   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1054   0         0.027     63         130.38    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1013   0         0.024     255        125.49    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╺━━━   994    0         0.027     255        123.02    0:00:08   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1012   0         0.021     31         123.53    0:00:08   0:00:09   
                                                  draws/s                       
  ━━━╸━━━   1067   0         0.027     255        130.30    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1024   0         0.024     127        125.14    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1008   0         0.025     319        123.18    0:00:08   0:00:09   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1025   0         0.021     95         123.52    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1080   0         0.027     127        130.12    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1040   0         0.024     63         125.29    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1023   0         0.025     31         123.33    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1039   0         0.021     255        123.43    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1094   0         0.027     159        130.12    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1056   0         0.024     63         125.62    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1039   0         0.025     255        123.59    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1051   0         0.021     191        123.35    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1112   0         0.027     31         130.57    0:00:08   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1071   0         0.024     255        125.79    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1053   0         0.025     255        123.63    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1062   0         0.021     255        122.98    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1125   0         0.027     255        130.26    0:00:08   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1084   0         0.024     255        125.56    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1067   0         0.025     255        123.63    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1079   0         0.021     63         123.26    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1143   0         0.027     255        130.63    0:00:08   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1096   0         0.024     127        125.37    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1081   0         0.025     63         123.59    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1088   0         0.021     31         122.75    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1153   0         0.027     63         130.08    0:00:08   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1106   0         0.024     127        124.76    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1088   0         0.025     255        122.89    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1106   0         0.021     95         123.22    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1168   0         0.027     127        130.41    0:00:08   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1120   0         0.024     191        124.84    0:00:08   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1100   0         0.025     255        122.67    0:00:08   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1118   0         0.021     127        123.16    0:00:09   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1184   0         0.027     127        130.44    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1134   0         0.024     127        125.05    0:00:09   0:00:08   
                                                  draws/s                       
  ━━━╸━━━   1111   0         0.025     255        122.44    0:00:09   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━╸━━━   1129   0         0.021     255        122.90    0:00:09   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1197   0         0.027     255        130.27    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1149   0         0.024     7          125.05    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1123   0         0.025     255        122.32    0:00:09   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1143   0         0.021     159        122.90    0:00:09   0:00:08   
                                                  draws/s                       
  ━━━━╺━━   1211   0         0.027     255        130.26    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1159   0         0.024     63         124.71    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━╸━━━   1137   0         0.025     63         122.34    0:00:09   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1157   0         0.021     127        122.88    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1223   0         0.027     191        129.89    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1177   0         0.024     31         125.11    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1154   0         0.025     63         122.68    0:00:09   0:00:08   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1166   0         0.021     511        122.43    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1243   0         0.027     127        130.47    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1191   0         0.024     63         125.08    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1165   0         0.025     127        122.43    0:00:09   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1176   0         0.021     255        122.03    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1257   0         0.027     127        130.46    0:00:09   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1206   0         0.024     95         125.19    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1175   0         0.025     255        121.99    0:00:09   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1188   0         0.021     255        121.92    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1267   0         0.027     191        129.98    0:00:09   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1220   0         0.024     255        125.24    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1193   0         0.025     127        122.40    0:00:09   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1206   0         0.021     15         122.28    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1278   0         0.027     255        129.67    0:00:09   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1235   0         0.024     95         125.27    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1204   0         0.025     255        122.15    0:00:09   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1218   0         0.021     255        122.10    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1290   0         0.027     159        129.35    0:00:09   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1248   0         0.024     127        125.15    0:00:09   0:00:07   
                                                  draws/s                       
  ━━━━╺━━   1218   0         0.025     127        122.18    0:00:09   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1229   0         0.021     255        121.92    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1303   0         0.027     127        129.26    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1265   0         0.024     27         125.52    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1229   0         0.025     127        121.98    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1241   0         0.021     63         121.73    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1318   0         0.027     127        129.29    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1277   0         0.024     239        125.24    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1243   0         0.025     127        121.96    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1253   0         0.021     255        121.53    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1331   0         0.027     239        129.19    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1291   0         0.024     31         125.23    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1259   0         0.025     127        122.21    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1263   0         0.021     255        121.21    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1344   0         0.027     127        128.99    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1303   0         0.024     127        125.05    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1274   0         0.025     127        122.25    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1275   0         0.021     31         121.05    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1359   0         0.027     127        129.01    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1315   0         0.024     191        124.83    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╺━━   1285   0         0.025     159        122.14    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╺━━   1284   0         0.021     127        120.60    0:00:10   0:00:07   
                                                  draws/s                       
  ━━━━╸━━   1367   0         0.027     127        128.49    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1325   0         0.024     95         124.52    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1294   0         0.025     255        121.61    0:00:10   0:00:07   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1298   0         0.021     255        120.74    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1377   0         0.027     255        128.12    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1338   0         0.024     383        124.49    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1311   0         0.025     215        121.93    0:00:10   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1320   0         0.021     159        121.45    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1386   0         0.027     383        127.56    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1354   0         0.024     127        124.62    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1322   0         0.025     255        121.73    0:00:10   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1334   0         0.021     95         121.63    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1400   0         0.027     255        127.59    0:00:10   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1364   0         0.024     191        124.38    0:00:10   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1335   0         0.025     63         121.70    0:00:10   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1346   0         0.021     127        121.45    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1409   0         0.027     255        127.20    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1376   0         0.024     191        124.36    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1344   0         0.025     127        121.46    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1359   0         0.021     63         121.44    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1418   0         0.027     255        126.76    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1389   0         0.024     15         124.25    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━╸━━   1358   0         0.025     255        121.39    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1373   0         0.021     127        121.43    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━━╺━   1431   0         0.027     127        126.68    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1399   0         0.024     127        123.82    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1374   0         0.025     63         121.58    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1388   0         0.021     127        121.51    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━━╺━   1446   0         0.027     255        126.66    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1413   0         0.024     255        123.77    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1388   0         0.025     127        121.58    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1400   0         0.021     127        121.40    0:00:11   0:00:06   
                                                  draws/s                       
  ━━━━━╺━   1460   0         0.027     127        126.59    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1424   0         0.024     127        123.58    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1399   0         0.025     511        121.34    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1414   0         0.021     127        121.54    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1473   0         0.027     255        126.49    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1436   0         0.024     255        123.41    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1408   0         0.025     127        120.99    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━╸━━   1427   0         0.021     63         121.35    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1482   0         0.027     231        126.11    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1447   0         0.024     255        123.17    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━╸━━   1420   0         0.025     191        120.85    0:00:11   0:00:06   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1441   0         0.021     127        121.43    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1497   0         0.027     191        126.19    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1466   0         0.024     63         123.55    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1436   0         0.025     159        121.10    0:00:11   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1450   0         0.021     127        121.13    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1510   0         0.027     255        126.07    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1482   0         0.024     31         123.76    0:00:11   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1448   0         0.025     255        121.04    0:00:11   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1460   0         0.021     223        120.81    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1517   0         0.027     127        125.59    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1491   0         0.024     127        123.51    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1454   0         0.025     63         120.38    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1471   0         0.021     255        120.68    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1530   0         0.027     191        125.47    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1506   0         0.024     63         123.52    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1468   0         0.025     63         120.44    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1481   0         0.021     255        120.41    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1551   0         0.027     95         126.08    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1516   0         0.024     95         123.24    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1480   0         0.025     71         120.39    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1491   0         0.021     127        120.14    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╺━   1561   0         0.027     63         125.76    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1532   0         0.024     71         123.44    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1489   0         0.025     31         120.09    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1502   0         0.021     127        119.98    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╸   1577   0         0.027     63         125.91    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1546   0         0.024     63         123.49    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1502   0         0.025     127        120.06    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1518   0         0.021     255        120.13    0:00:12   0:00:05   
                                                  draws/s                       
  ━━━━━╸   1588   0         0.027     255        125.72    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1560   0         0.024     255        123.54    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1516   0         0.025     143        120.05    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1530   0         0.021     159        120.08    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1596   0         0.027     255        125.33    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1574   0         0.024     159        123.55    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1528   0         0.025     127        119.93    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1544   0         0.021     31         120.10    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1613   0         0.027     31         125.50    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1587   0         0.024     255        123.48    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1542   0         0.025     127        120.01    0:00:12   0:00:05   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1555   0         0.021     63         119.95    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1626   0         0.027     127        125.44    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1599   0         0.024     111        123.37    0:00:12   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1556   0         0.025     255        120.09    0:00:12   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╺━   1568   0         0.021     191        119.99    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1638   0         0.027     119        125.37    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1612   0         0.024     127        123.43    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╺━   1568   0         0.025     95         120.01    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1575   0         0.021     255        119.70    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1653   0         0.027     31         125.52    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1626   0         0.024     191        123.46    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1577   0         0.025     255        119.80    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1586   0         0.021     127        119.35    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1664   0         0.027     511        125.26    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1644   0         0.024     63         123.76    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1592   0         0.025     255        119.96    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1595   0         0.021     127        119.16    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1673   0         0.027     255        124.92    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1659   0         0.024     75         123.88    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1601   0         0.025     63         119.60    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1601   0         0.021     31         118.65    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1682   0         0.027     63         124.55    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1665   0         0.024     127        123.35    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1607   0         0.025     63         119.06    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1611   0         0.021     63         118.39    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1695   0         0.027     127        124.59    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1679   0         0.024     255        123.38    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1619   0         0.025     127        118.99    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1625   0         0.021     127        118.41    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━╸   1709   0         0.027     95         124.53    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1686   0         0.024     127        122.97    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1632   0         0.025     127        118.94    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1633   0         0.021     23         118.10    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━━   1719   0         0.027     383        124.30    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1694   0         0.024     255        122.55    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1641   0         0.025     127        118.64    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1642   0         0.021     255        117.78    0:00:13   0:00:04   
                                                  draws/s                       
  ━━━━━━   1730   0         0.027     191        124.13    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1700   0         0.024     63         122.04    0:00:13   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1649   0         0.025     255        118.37    0:00:13   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1650   0         0.021     127        117.50    0:00:14   0:00:04   
                                                  draws/s                       
  ━━━━━━   1740   0         0.027     223        123.82    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1709   0         0.024     143        121.68    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1659   0         0.025     63         118.15    0:00:14   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1660   0         0.021     127        117.26    0:00:14   0:00:04   
                                                  draws/s                       
  ━━━━━━   1748   0         0.027     255        123.51    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1722   0         0.024     127        121.63    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1672   0         0.025     111        118.14    0:00:14   0:00:04   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1671   0         0.021     255        117.09    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1763   0         0.027     191        123.56    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1733   0         0.024     63         121.46    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1684   0         0.025     255        118.10    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1678   0         0.021     383        116.77    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1771   0         0.027     319        123.17    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1741   0         0.024     127        121.29    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1690   0         0.025     511        117.62    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1686   0         0.021     255        116.35    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1782   0         0.027     127        122.99    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1749   0         0.024     127        120.73    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1700   0         0.025     71         117.48    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1696   0         0.021     191        116.16    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1794   0         0.027     255        122.81    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1762   0         0.024     255        120.65    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━╸   1712   0         0.025     191        117.32    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1702   0         0.021     191        115.93    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1801   0         0.027     127        122.73    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1768   0         0.024     255        120.45    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1718   0         0.025     239        116.76    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━╸   1712   0         0.021     255        115.36    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1815   0         0.027     255        122.32    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1779   0         0.024     127        120.04    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1729   0         0.025     255        116.53    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1723   0         0.021     127        115.25    0:00:14   0:00:03   
                                                  draws/s                       
  ━━━━━━   1827   0         0.027     31         122.21    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1790   0         0.024     255        119.78    0:00:14   0:00:02   
                                                  draws/s                       
  ━━━━━━   1741   0         0.025     63         116.56    0:00:14   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1734   0         0.021     31         115.13    0:00:15   0:00:03   
                                                  draws/s                       
  ━━━━━━   1840   0         0.027     127        122.18    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1805   0         0.024     191        119.86    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1758   0         0.025     63         116.81    0:00:15   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1745   0         0.021     63         115.02    0:00:15   0:00:03   
                                                  draws/s                       
  ━━━━━━   1850   0         0.027     351        121.96    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1816   0         0.024     127        119.81    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1768   0         0.025     63         116.57    0:00:15   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1758   0         0.021     31         115.04    0:00:15   0:00:03   
                                                  draws/s                       
  ━━━━━━╸   1861   0         0.027     127        121.87    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1828   0         0.024     95         119.65    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1775   0         0.025     255        116.22    0:00:15   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1771   0         0.021     95         115.08    0:00:15   0:00:03   
                                                  draws/s                       
  ━━━━━━╸   1873   0         0.027     255        121.79    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1842   0         0.024     255        119.71    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1781   0         0.025     511        115.82    0:00:15   0:00:03   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1783   0         0.021     255        115.10    0:00:15   0:00:03   
                                                  draws/s                       
  ━━━━━━╸   1883   0         0.027     255        121.47    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1854   0         0.024     255        119.64    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1788   0         0.025     215        115.38    0:00:15   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1796   0         0.021     255        115.08    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1895   0         0.027     95         121.49    0:00:15   0:00:01   
                                                  draws/s                       
  ━━━━━━╸   1863   0         0.024     255        119.40    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1796   0         0.025     127        115.06    0:00:15   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1809   0         0.021     31         115.09    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1905   0         0.027     255        121.21    0:00:15   0:00:01   
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  ━━━━━━╸   1872   0         0.024     63         119.18    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━   1807   0         0.025     255        114.99    0:00:15   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1819   0         0.021     127        114.88    0:00:15   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1913   0         0.027     255        121.13    0:00:15   0:00:01   
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  ━━━━━━╸   1877   0         0.024     255        118.86    0:00:15   0:00:02   
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  ━━━━━━   1814   0         0.025     127        114.94    0:00:15   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1831   0         0.021     127        114.76    0:00:15   0:00:02   
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  ━━━━━━╸   1927   0         0.027     127        120.79    0:00:15   0:00:01   
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  ━━━━━━╸   1893   0         0.024     63         118.69    0:00:15   0:00:01   
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  ━━━━━━   1828   0         0.025     255        114.59    0:00:15   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1842   0         0.021     255        114.67    0:00:16   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1938   0         0.027     127        120.68    0:00:16   0:00:01   
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  ━━━━━━╸   1910   0         0.024     63         118.96    0:00:16   0:00:01   
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  ━━━━━━   1837   0         0.025     127        114.42    0:00:16   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━   1852   0         0.021     63         114.46    0:00:16   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1947   0         0.027     191        120.46    0:00:16   0:00:01   
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  ━━━━━━╸   1926   0         0.024     63         119.07    0:00:16   0:00:01   
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  ━━━━━━   1848   0         0.025     255        114.25    0:00:16   0:00:02   
                                                  draws/s                       
                                                                                
                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1863   0         0.021     319        114.42    0:00:16   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1959   0         0.027     127        120.27    0:00:16   0:00:01   
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  ━━━━━━╸   1940   0         0.024     127        119.11    0:00:16   0:00:01   
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  ━━━━━━╸   1859   0         0.025     127        114.15    0:00:16   0:00:02   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1873   0         0.021     255        114.19    0:00:16   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1970   0         0.027     255        120.13    0:00:16   0:00:01   
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  ━━━━━━╸   1952   0         0.024     63         119.04    0:00:16   0:00:01   
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  ━━━━━━╸   1871   0         0.025     31         114.11    0:00:16   0:00:02   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1882   0         0.021     255        114.05    0:00:16   0:00:02   
                                                  draws/s                       
  ━━━━━━╸   1985   0         0.027     127        120.24    0:00:16   0:00:01   
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  ━━━━━━╸   1965   0         0.024     127        119.05    0:00:16   0:00:01   
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  ━━━━━━╸   1880   0         0.025     63         114.01    0:00:16   0:00:02   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1897   0         0.021     95         114.16    0:00:16   0:00:01   
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  ━━━━━━╸   1997   0         0.027     255        120.19    0:00:16   0:00:01   
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  ━━━━━━╸   1977   0         0.024     95         118.96    0:00:16   0:00:01   
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  ━━━━━━╸   1892   0         0.025     127        113.93    0:00:16   0:00:02   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1904   0         0.021     31         114.32    0:00:16   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━╸   1979   0         0.024     255        118.92    0:00:16   0:00:01   
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  ━━━━━━╸   1896   0         0.025     23         113.87    0:00:16   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1912   0         0.021     255        114.25    0:00:16   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━╸   1988   0         0.024     255        118.86    0:00:16   0:00:01   
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  ━━━━━━╸   1907   0         0.025     63         113.99    0:00:16   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1921   0         0.021     127        114.27    0:00:16   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━╸   1994   0         0.024     223        118.63    0:00:16   0:00:01   
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  ━━━━━━╸   1917   0         0.025     207        113.82    0:00:16   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1927   0         0.021     127        114.12    0:00:16   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1923   0         0.025     127        113.93    0:00:16   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1936   0         0.021     255        114.11    0:00:16   0:00:01   
                                                  draws/s                       
  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1935   0         0.025     127        114.08    0:00:16   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1949   0         0.021     127        114.14    0:00:17   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
                                                  draws/s                       
  ━━━━━━╸   1951   0         0.025     63         114.36    0:00:17   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1967   0         0.021     127        114.47    0:00:17   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1966   0         0.025     255        114.47    0:00:17   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1985   0         0.021     75         114.80    0:00:17   0:00:01   
                                                  draws/s                       
  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1978   0         0.025     79         114.43    0:00:17   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━╸   1996   0         0.021     255        114.77    0:00:17   0:00:01   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1993   0         0.025     127        114.58    0:00:17   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━━   2000   0         0.021     63         114.78    0:00:17   0:00:00   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━╸   1995   0         0.025     255        114.58    0:00:17   0:00:01   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━━   2000   0         0.021     63         114.78    0:00:17   0:00:00   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.025     31         114.65    0:00:17   0:00:00   
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                             Step      Grad                                     
  Progre…   Draw   Diverg…   size      evals      Speed     Elapsed   Remaini…  
 ────────────────────────────────────────────────────────────────────────────── 
  ━━━━━━━   2000   0         0.021     63         114.78    0:00:17   0:00:00   
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  ━━━━━━━   2000   0         0.027     127        120.07    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.024     63         118.40    0:00:16   0:00:00   
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  ━━━━━━━   2000   0         0.025     31         114.65    0:00:17   0:00:00   
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?25hSampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 18 seconds.
Sampling: [beta, y_hat, y_hat_sigma]
Sampling: [y_hat]
Sampling: [y_hat]
Sampling: [y_hat]
Sampling: [y_hat]
fig, ax = result.plot(plot_predictors=True)
../_images/556a96fdc6255f3ffdb61422ec9dc42c57a6c3fdac760d04a0c38d07981062a5.png
result.summary()
================================SyntheticControl================================
Control units: ['a', 'b', 'c', 'd', 'e', 'f', 'g']
Treated unit: actual
Model coefficients:
    a            0.054, 94% HDI [0.0021, 0.13]
    b            0.2, 94% HDI [0.14, 0.25]
    c            0.21, 94% HDI [0.12, 0.29]
    d            0.061, 94% HDI [0.0072, 0.13]
    e            0.26, 94% HDI [0.22, 0.29]
    f            0.11, 94% HDI [0.071, 0.15]
    g            0.11, 94% HDI [0.075, 0.15]
    y_hat_sigma  0.25, 94% HDI [0.21, 0.3]
Pre-treatment correlation (actual): 0.9992

We can get nicely formatted tables from our integration with the maketables package.

from maketables import ETable

result.set_maketables_options(hdi_prob=0.95)
ETable(result, coef_fmt="b:.3f \n [ci95l:.3f, ci95u:.3f]")
y
(1)
coef
a 0.054
[0.000, 0.121]
b 0.196
[0.144, 0.253]
c 0.209
[0.126, 0.295]
d 0.061
[0.000, 0.120]
e 0.256
[0.224, 0.290]
f 0.112
[0.071, 0.153]
g 0.113
[0.073, 0.151]
stats
N 100
Bayesian R2 0.998
Format of coefficient cell: Coefficient [95% CI Lower, 95% CI Upper]

As well as the model coefficients, we might be interested in the average causal impact and average cumulative causal impact.

az.summary(result.post_impact.mean("obs_ind"))
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
x[actual] -1.239 0.363 -1.906 -0.585 0.012 0.006 946.0 1492.0 1.0

Warning

Care must be taken with the mean impact statistic. It only makes sense to use this statistic if it looks like the intervention had a lasting (and roughly constant) effect on the outcome variable. If the effect is transient, then clearly there will be a lot of post-intervention period where the impact of the intervention has ‘worn off’. If so, then it will be hard to interpret the mean impacts real meaning.

We can also ask for the summary statistics of the cumulative causal impact.

# get index of the final time point
index = result.post_impact_cumulative.obs_ind.max()
# grab the posterior distribution of the cumulative impact at this final time point
last_cumulative_estimate = result.post_impact_cumulative.sel({"obs_ind": index})
# get summary stats
az.summary(last_cumulative_estimate)
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
x[actual] -37.181 10.888 -57.189 -17.546 0.352 0.182 946.0 1492.0 1.0

Effect Summary Reporting#

For decision-making, you often need a concise summary of the causal effect with key statistics. The effect_summary() method provides a decision-ready report with average and cumulative effects, HDI intervals, tail probabilities, and relative effects.

# Generate effect summary for the full post-period
stats = result.effect_summary(treated_unit="actual")
stats.table
mean median hdi_lower hdi_upper p_gt_0 relative_mean relative_hdi_lower relative_hdi_upper
average -1.239357 -1.241476 -1.913526 -0.545315 0.0 -5.049057 -7.617849 -2.295991
cumulative -37.180701 -37.244276 -57.405766 -16.359464 0.0 -5.049057 -7.617849 -2.295991
print(stats.text)
During the Post-period (70 to 99), the response variable had an average value of approx. 23.21. By contrast, in the absence of an intervention, we would have expected an average response of 24.44. The 95% interval of this counterfactual prediction is [23.75, 25.12]. Subtracting this prediction from the observed response yields an estimate of the causal effect the intervention had on the response variable. This effect is -1.24 with a 95% interval of [-1.91, -0.55].

Summing up the individual data points during the Post-period, the response variable had an overall value of 696.16. By contrast, had the intervention not taken place, we would have expected a sum of 733.34. The 95% interval of this prediction is [712.52, 753.57].

The 95% HDI of the effect [-1.91, -0.55] does not include zero. The posterior probability of a decrease is 1.000. Relative to the counterfactual, the effect represents a -5.05% change (95% HDI [-7.62%, -2.30%]).

This analysis assumes that the control units used to construct the synthetic counterfactual were not themselves affected by the intervention, and that the pre-treatment relationship between control and treated units remains stable throughout the post-treatment period. We recommend inspecting model fit, examining pre-intervention trends, and conducting sensitivity analyses (e.g., placebo tests) to support any causal conclusions drawn from this analysis.

You can customize the summary in several ways:

  • Window: Analyze a specific time period instead of the full post-period

  • Direction: Specify whether you’re testing for an increase, decrease, or two-sided effect

  • Options: Include/exclude cumulative or relative effects

# Example: Analyze first half of post-period
post_indices = result.datapost.index
window_start = post_indices[0]
window_end = post_indices[len(post_indices) // 2]

stats_windowed = result.effect_summary(
    window=(window_start, window_end),
    treated_unit="actual",
    direction="two-sided",
    cumulative=True,
    relative=True,
)

stats_windowed.table
mean median hdi_lower hdi_upper p_two_sided prob_of_effect relative_mean relative_hdi_lower relative_hdi_upper
average -2.655351 -2.641739 -3.146364 -2.144796 0.0 1.0 -11.823094 -13.782409 -9.845488
cumulative -42.485612 -42.267822 -50.341832 -34.316743 0.0 1.0 -11.823094 -13.782409 -9.845488
print(stats_windowed.text)
During the Post-period (70 to 85), the response variable had an average value of approx. 19.78. By contrast, in the absence of an intervention, we would have expected an average response of 22.44. The 95% interval of this counterfactual prediction is [21.93, 22.93]. Subtracting this prediction from the observed response yields an estimate of the causal effect the intervention had on the response variable. This effect is -2.66 with a 95% interval of [-3.15, -2.14].

Summing up the individual data points during the Post-period, the response variable had an overall value of 316.49. By contrast, had the intervention not taken place, we would have expected a sum of 358.98. The 95% interval of this prediction is [350.81, 366.84].

The 95% HDI of the effect [-3.15, -2.14] does not include zero. The posterior probability of an effect is 1.000. Relative to the counterfactual, the effect represents a -11.82% change (95% HDI [-13.78%, -9.85%]).

This analysis assumes that the control units used to construct the synthetic counterfactual were not themselves affected by the intervention, and that the pre-treatment relationship between control and treated units remains stable throughout the post-treatment period. We recommend inspecting model fit, examining pre-intervention trends, and conducting sensitivity analyses (e.g., placebo tests) to support any causal conclusions drawn from this analysis.