Posterior Sampling and Summarization

Summary

Chapter 3 of Statistical Rethinking shows how to work with posterior distributions using samples. Three tasks: (1) summarizing with intervals and point estimates, (2) simulating predictions via the posterior predictive distribution, and (3) model checking through posterior predictive checks.

Working with Samples

The fundamental tool: draw samples from the posterior, then manipulate those samples. This transforms integral calculus into data summary:

• Probability of $p<0.5$? → count samples below 0.5
• 89% credible interval? → find quantiles of samples
• Expected prediction? → simulate data for each sample

Summarizing the Posterior

Intervals

Type	Definition	Property
Percentile interval (PI)	Central quantiles (e.g., 5.5% and 94.5%)	Equal mass in each tail
Highest posterior density interval (HPDI)	Narrowest interval containing X% of mass	Always includes the mode

When PI and HPDI Disagree

If the two intervals differ substantially, don’t rely on either — plot the entire posterior instead. The posterior is the estimate.

Point Estimates and Loss Functions

Different loss functions imply different point estimates:

• Absolute loss $|d-p|$ → median minimizes expected loss
• Quadratic loss $(d-p{)}^2$ → mean minimizes expected loss
• Zero-one loss → mode (MAP) minimizes expected loss

McElreath’s key insight: you rarely need a point estimate. The entire posterior distribution is the Bayesian answer.

Posterior Predictive Distribution

Two sources of uncertainty in predictions:

1. Parameter uncertainty — the posterior distribution over $\theta$
2. Observation uncertainty — the sampling process given $\theta$

The posterior predictive distribution integrates over both:

$$p(y^{\text{new}}|y)=\int p(y^{\text{new}}|\theta )p(\theta |y)d\theta$$

In practice: for each posterior sample of $\theta$, simulate an observation → the collection of simulated observations is the posterior predictive distribution.

Model Checking

Use posterior predictive checks to assess model adequacy — compare simulated data to observed data on various dimensions. If the model fits well, simulated data should “look like” the real data.

A model can fit the observed summary (e.g., total count) while failing on other aspects (e.g., longest run, number of switches). Always check multiple summary statistics.

See Also

• Model Checking — BDA3’s formal treatment of posterior predictive checks (Ch 6)
• Decision Analysis — BDA3’s treatment of loss functions and optimal decisions
• Garden of Forking Data — building the posterior that this chapter summarizes
• Bayesian Workflow - Overview — the iterative cycle of fitting, checking, and revising
• Statistical Rethinking - Overview