Decision Analysis

Summary

Chapter 9 of BDA3 connects Bayesian inference to decision-making. The optimal decision minimizes expected loss (or maximizes expected utility) under the posterior distribution.

Framework

Given a decision $d$, unknown parameters $\theta$, and a loss function $L(d,\theta )$:

$$d^{\ast }=\arg \underset{d}{\min }\text{E}[L(d,\theta )\mid y]=\arg \underset{d}{\min }\int L(d,\theta )p(\theta \mid y)d\theta$$

Common loss functions yield familiar estimators:

• Squared error loss → posterior mean
• Absolute error loss → posterior median
• 0-1 loss → posterior mode

Applications Covered

• Survey incentives: using regression predictions to optimize survey response rates
• Medical screening: multistage decision making under uncertainty
• Home radon: hierarchical model informing household-level decisions
• Personal vs. institutional decisions: different utility functions for individual vs. policy decisions

Key Insight

Tip

The full posterior distribution — not just point estimates — flows directly into decisions. This is a major advantage of the Bayesian approach: uncertainty quantification naturally informs the cost of being wrong.

See Also

• Probability and Bayesian Inference — the posterior that feeds into decisions
• Hierarchical Models — partial pooling often improves decisions by reducing variance
• Model Comparison — choosing between models before making decisions
• Overfitting and Information Criteria — model selection criteria that inform which posterior to use
• Counterfactual Inference — counterfactual thinking as a prerequisite for decision framing
• Model Checking — a model must pass posterior predictive checks before its posterior can be trusted in a decision analysis