Highest Posterior Density Region and Central Credible Region

Question

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

Highest Posterior Density Region:

The Highest Posterior Density Region

Answer 1

I stumbled across this post trying to find a way to estimate an HDI from an MCMC sample but none of the answers worked for me. Like aloctavodia, I adapted an R example from the book Doing Bayesian Data Analysis to Python. I needed to compute a 95% HDI from an MCMC sample. Here's my solution:

import numpy as np
def HDI_from_MCMC(posterior_samples, credible_mass):
    # Computes highest density interval from a sample of representative values,
    # estimated as the shortest credible interval
    # Takes Arguments posterior_samples (samples from posterior) and credible mass (normally .95)
    sorted_points = sorted(posterior_samples)
    ciIdxInc = np.ceil(credible_mass * len(sorted_points)).astype('int')
    nCIs = len(sorted_points) - ciIdxInc
    ciWidth = [0]*nCIs
    for i in range(0, nCIs):
    ciWidth[i] = sorted_points[i + ciIdxInc] - sorted_points[i]
    HDImin = sorted_points[ciWidth.index(min(ciWidth))]
    HDImax = sorted_points[ciWidth.index(min(ciWidth))+ciIdxInc]
    return(HDImin, HDImax)

The method above is giving me logical answers based on the data I have!