Summarizing Posterior Inference#

Part 1 · Stage 2 · 📍 Single-Parameter Models & Priors · Lesson 013 of 144 · beginner

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Important

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The posterior is the answer#

Formally, the whole answer of a Bayesian analysis is the posterior distribution. But a distribution is hard to report and harder to act on, so we summarise it — carefully, because every summary discards information, and some discard the wrong information.

Point summaries#

Three point estimates are standard, and they differ once the posterior is skewed:

the posterior mean \(\mathrm{E}[\theta \mid y]\) — the Bayes estimator under squared-error loss, and the usual default;
the posterior median — robust, and optimal under absolute-error loss;
the posterior mode (the MAP estimate) — the density’s peak, and the point where Bayesian estimation coincides with penalised maximum likelihood.

For a symmetric unimodal posterior all three agree. When they disagree, the disagreement itself is information: the posterior is asymmetric, and no single number represents it.

Intervals: two kinds#

A credible interval is any region holding a stated posterior probability — say 95%. Two constructions dominate:

the equal-tailed interval (ETI), cutting 2.5% from each tail (the 2.5th and 97.5th percentiles);
the highest-density interval (HDI/HPD), the shortest interval containing 95% of the mass — equivalently, every point inside has higher density than every point outside.

They coincide for symmetric posteriors. For skewed ones they do not, and the ETI can include parameter values less plausible than some it excludes; the HDI always contains the mode.

import numpy as np, arviz as az
draws = ...                                  # posterior draws
draws.mean(), np.median(draws)               # point summaries
np.percentile(draws, [2.5, 97.5])            # equal-tailed 95% CI
az.hdi(draws, hdi_prob=0.95)                 # highest-density 95% CI
(draws > 0.5).mean()                         # P(theta > 0.5 | y)

Say what you mean#

The interpretive prize is real: a 95% credible interval supports the statement “there is a 95% probability :math:`theta` lies in this range” — precisely what a frequentist confidence interval does not license. But do not let the summary replace the distribution: plot the posterior, and report the probability of the hypotheses you actually care about, which the draws give directly.

Hint