Bayesian decision theory in diﬀerent contexts#

Part 2 · Stage 7 · 🗳️ Data Collection & Decisions · Lesson 057 of 144 · intermediate

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Important

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From inference to action#

A posterior is not a decision. To act you need one more ingredient: a utility function stating what outcomes are worth. Bayesian decision theory then supplies the rule — among the available actions, choose the one whose utility is greatest on average over the posterior.

The rule#

Let \(a\) be an action, \(\theta\) the unknown state, and \(U(a, \theta)\) the utility of taking \(a\) when the truth is \(\theta\). The optimal action maximises expected utility:

\[a^{*} = \arg\max_{a} \; \mathrm{E}_{\theta \mid y}\bigl[U(a, \theta)\bigr] = \arg\max_{a} \int U(a, \theta) \; p(\theta \mid y) \; d\theta .\]

With posterior draws this is a one-line computation — the expectation becomes an average, and the optimisation a search over the (usually few) candidate actions.

import numpy as np
theta = idata.posterior["theta"].values.ravel()      # posterior draws
actions = np.linspace(0, 1, 101)                     # candidate actions
EU = [np.mean(utility(a, theta)) for a in actions]   # expected utility per action
best = actions[int(np.argmax(EU))]

Note what the posterior does here: it weights each possible truth by its plausibility. The full distribution matters, not just its mean — because utilities are usually nonlinear, and \(\mathrm{E}[U(a, \theta)] \ne U(a, \mathrm{E}[\theta])\).

Why estimation is not decision#

Familiar Bayesian summaries are decisions in disguise, each optimal under a particular loss (negative utility). Squared-error loss gives the posterior mean; absolute-error loss gives the median; 0–1 loss gives the mode. So the choice of point estimate is not a matter of taste but of the cost of being wrong — and when those costs are asymmetric (a flood barrier too low versus too high) the optimal action can lie far out in a tail, nowhere near any conventional estimate.

Contexts differ#

Three settings recur through this stage, and their utilities differ in kind. A one-shot choice: pick \(a\) once, as with a survey incentive. A sequential problem: act, observe, act again, where an early action buys information for later ones. And a hierarchical problem, where decisions are made for many units at once and shrinkage governs each. The final lesson adds the sharpest distinction of all — whose utility, the individual’s or the institution’s, since the two can rationally disagree.

Hint