Reporting Bayesian results

Bayesian Data Analysis in Python

Michal Oleszak

Machine Learning Engineer

The honest way

Report the prior and the posterior of each parameter

posterior_draws

array([8.02800413, 8.97359548, 7.57437476, ..., 5.85264609, 7.92875104,
       7.41463758])

Plot prior and posterior distributions

sns.kdeplot(prior_draws, shade=True, label="prior")
sns.kdeplot(posterior_draws, shade=True, label="posterior")

The honest way

Two density plots overlaid onto each other. Both are bell-shaped. Prior is centered at 0. short and wide, posterior is centered around 7 and is much taller and narrower.

The honest way

Two density plots overlaid onto each other. Both are bell-shaped. Prior is centered at 0. short and wide, posterior is centered around 7 and is much taller and narrower.

The honest way

Two density plots overlaid onto each other. Both are bell-shaped. Prior is centered at 0. short and wide, posterior is centered around 7 and is much taller and narrower.

The honest way

Two density plots overlaid onto each other. Both are bell-shaped. Prior is centered at 0. short and wide, posterior is centered around 7 and is much taller and narrower.

Bayesian point estimates

No single number can fully convey the complete information contained in a distribution
However, sometimes a point estimate of a parameter is needed

A bell-shaped density plot.

Bayesian point estimates

No single number can fully convey the complete information contained in a distribution
However, sometimes a point estimate of a parameter is needed

posterior_mean = np.mean(posterior_draws)

A bell-shaped density plot with a vertical line marking its mean.

Bayesian point estimates

No single number can fully convey the complete information contained in a distribution
However, sometimes a point estimate of a parameter is needed

posterior_mean = np.mean(posterior_draws)
posterior_median = np.median(posterior_draws)

A bell-shaped density plot with a vertical line marking its median.

Bayesian point estimates

No single number can fully convey the complete information contained in a distribution
However, sometimes a point estimate of a parameter is needed

posterior_mean = np.mean(posterior_draws)
posterior_median = np.median(posterior_draws)
posterior_p75 = np.percentile(posterior_draws, 75)

A bell-shaped density plot with a vertical line marking its 75th percentile.

Credible intervals

Such an interval that the probability that the parameter falls inside it is x%
The wider the credible interval, the more uncertainty in parameter estimate
Parameter is random, so it can fall into an interval with some probability
In the frequentist world, the (confidence) interval is random while the parameter is fixed

A bell-shaped density plot with two dashed vertical lines marking the region between them.

Highest Posterior Density (HPD)

A GIF of a bell-shaped density plot showing a horizontal line descending down the plot area, resulting in an increasing are of the probability mass located between the points, where the line intersects with the plot.

import arviz as az

hpd = az.hdi(posterior_draws, 
             hdi_prob=0.9)
print(hpd)

[-4.86840193  4.96075498]

Let's practice reporting Bayesian results!

Bayesian Data Analysis in Python