Chapter 4

Chapter 4: Bayesian Estimators and Credible Intervals

The posterior distribution is the complete result of our inference. However, we often need to summarize it with a single point estimate or an interval.

4.1 Bayesian Point Estimators

Posterior Mean: The average of the posterior distribution. This is the most common point estimate.
Posterior Median: The 50th percentile of the posterior. More robust to skewed posteriors.
Maximum a Posteriori (MAP): The peak (mode) of the posterior. Can be found with optimization, but it ignores the shape of the distribution and can sometimes be misleading.

4.2 Credible Intervals

A credible interval is a range that contains the parameter with a specific posterior probability. An X% credible interval has a direct, intuitive interpretation: "Given the data, there is an X% probability that the true value of the parameter lies within this interval."
Equal-Tailed Interval (ETI): Formed by taking the quantiles (e.g., the 2.5% and 97.5% quantiles for a 95% interval).
Highest Posterior Density (HPD) Interval: The narrowest possible interval containing the specified probability.

4.3 Code Example: Summarizing the Posterior

We can easily compute these summaries from our MCMC trace using the arviz library.
Python

# Continuing from the previous chapter's code...

# Use arviz to get a summary table
summary = az.summary(trace, var_names=['p'])
print(summary)

# Extract specific values
posterior_mean = summary['mean'].values[0]
posterior_median = summary['median'].values[0]
credible_interval = summary[['hdi_3%', 'hdi_97%']].values[0] # ArviZ provides HPD by default

print(f"\nPosterior Mean: {posterior_mean:.3f}")
print(f"Posterior Median: {posterior_median:.3f}")
print(f"94% Highest Density Interval (HDI): [{credible_interval[0]:.3f}, {credible_interval[1]:.3f}]")