Bayesian decision theory in context

Up to now, we’ve mostly done this workflow:

  1. specify a model,
  2. get the posterior $p(\theta \mid \text{data})$,
  3. maybe get a posterior predictive distribution.

Then we stop.

But in real life the reason we estimate “probability of cancer,” or “effect of treatment,” or “chance a state votes Democrat” is so that someone can make a decision: treat/not treat, run/not run the campaign, pay/not pay an incentive, etc.

Bayesian decision theory says: take your posterior and push it one step further—combine it with utilities/costs and pick the action that maximizes expected utility.

Two ways Bayesian inference shows up in decisions

  1. To get the predictive inputs for the decision.
    • A decision often depends on things we don’t know exactly: probability of recovery under treatment A, expected cost under policy B, etc.
    • Those things depend on unknown parameters (regression coefficients, population proportions).
    • We do Bayesian inference to get the posterior over those parameters → which gives us the posterior (predictive) distribution of the outcomes we care about.
  2. To update inside a multistage decision.
    • Sometimes we first do something (e.g. run a pilot, buy information) and then decide again.
    • After that first stage we observe new data, so we update the posterior again conditional on the earlier decision.
    • That’s the setup for valuing information and for decision trees.

So: Bayesian updating is used both before and inside the decision tree.

The 4-step template for Bayesian decision analysis

They give a very standard 4-step recipe:

  1. List decisions and outcomes.
    • Decisions: $d \in \mathcal{D}$ (e.g. choose incentive \$0, \$2, \$5; choose treatment A or B).
    • Outcomes: $x$ (can be money, lives, response rate, or a vector like $(\text{cost}, \text{lives})$).
    • Outcomes can include predicted future data $\tilde y$​ and even parameters θ\thetaθ if those matter.
  2. Get the distribution of outcomes for each decision.
    • For every ddd, you want $p(x \mid d)$.
    • In Bayesian terms: you condition on the decision, you don’t put a prior on d.
    • This is where you use your posterior/model to produce the predictive distribution under that decision.
  3. Specify a utility function $U(x)$.
    • Maps outcome(s) → real number.
    • If x is 1D (profit, life-years), U can be just that.
    • If x has multiple components, U has to encode trade-offs (e.g. dollars vs response rate, or dollars vs lives).
  4. Pick the decision with the highest expected utility.
    • Compute $\text{EU}(d) = E[U(x) \mid d] = \int U(x)\, p(x \mid d)\,dx$
    • Choose $d^\* = \arg\max_d \text{EU}(d)$.
    • In a multi-stage decision tree, you repeat this backward from the end, conditioning on whatever has been observed so far.

That’s the formal definition of Bayesian decision analysis.

Why it looks harder than plain inference

  • Plain Bayesian inference: average over uncertainty.
  • Bayesian decision analysis: average over uncertainty and optimize over actions.
  • That’s why decision trees feel busier: they have nodes for “chance” (where you integrate) and nodes for “decision” (where you maximize).

In practice, many analyses stop after step 2: they give the posterior or the predicted consequences of each option, and the actual decision-maker supplies the utility.

Summaries and model selection as decisions

They add a useful side note:

  • Sometimes we can’t or don’t want to show the whole posterior distribution (reporting rules, dashboards, management).
  • Then choosing what summary to report (point estimate, interval, predictive distribution) can itself be viewed as a decision problem, with a scoring rule as the utility.
    • Log score → best to report the whole predictive distribution.
    • Squared error → best to report posterior mean.
    • Absolute error → best to report posterior median.
  • Similarly, model selection can be seen as a decision:
    • Better predictive performance from a bigger model gives benefit,
    • but collecting extra variables/covariates costs money,
    • so you choose the model that maximizes expected utility = benefit – cost.

So the same logic (enumerate options → get predictive distribution → define utility → choose) applies to:

  • medical treatments,
  • survey incentives,
  • and even “which model should I use?”

Where the later sections go

They say:

  • An applied example where they compare options and their expected consequences but don’t fully optimize.
  • Multi-stage, with value of information.
  • A full decision analysis with utility maximization (dollars + lives).

So this section is the general framework; the details come in the examples.

In short: Bayesian inference gives you $p(x \mid d)$. Bayesian decision theory tells you to combine that with $U(x)$ and pick the $d$ with the largest $E[U(x)\mid d]$.