Highest Posterior Density Region and Central Credible Region

Given a posterior p(Θ|D) over some parameters Θ, one can define the following:

Highest Posterior Density Region:

The Highest Posterior Density Region is the set of most probable values of Θ that, in total, constitute 100(1-α) % of the posterior mass.

In other words, for a given α, we look for a p* that satisfies:

and then obtain the Highest Posterior Density Region as the set:

Central Credible Region:

Using the same notation as above, a Credible Region (or interval) is defined as:

Depending on the distribution, there could be many such intervals. The central credible interval is defined as a credible interval where there is (1-α)/2 mass on each tail.

Computation:

• For general distributions, given samples from the distribution, are there any built-ins in to obtain the two quantities above in Python or PyMC?

• For common parametric distributions (e.g. Beta, Gaussian, etc.) are there any built-ins or libraries to compute this using SciPy or statsmodels?

To calculate HPD you can leverage pymc3, Here is an example

import pymc3
from scipy.stats import norm
a = norm.rvs(size=10000)
pymc3.stats.hpd(a)