Posterior loss and Bayes decisions

For action $a$ and unknown state $\theta$, specify a loss $L(a,\theta)$ in consistent units. After observing $y$, the posterior risk is its expectation under $\pi(\theta\mid y)$. Choose an available action minimizing that expectation, assuming the expectations exist and a minimizer is available. With finitely many actions and finite losses this is direct comparison. With continuous actions, existence and constraints need attention; a differentiable stationary point need not be a global minimizer.

If an action changes future outcomes, the relevant loss may depend on a future observation as well as the unknown parameter. Integrate over the posterior predictive distribution conditional on that action, not over an observational distribution that silently assumes no intervention. An association between a chosen action and an outcome does not automatically identify its causal effect. The mathematical optimization can be flawless while the action-dependent model is wrong. State what intervention or stability assumptions permit transporting past information into this decision.

For squared error and a posterior with finite second moment, expand around the posterior mean. The cross term vanishes because the centered parameter has posterior expectation zero. Risk becomes a constant posterior variance plus squared distance from that mean, proving that the mean minimizes risk over all real actions. This derivation also shows why reporting only the mean hides uncertainty: two posteriors can have the same minimizing action and very different unavoidable posterior risks.

Absolute-error loss instead admits posterior medians as minimizers when the first absolute moment is finite. Asymmetric linear loss selects a posterior quantile, with its level determined by the relative costs of overestimation and underestimation. For a continuous parameter, zero-one loss for exact equality makes every action have essentially the same loss under a continuous posterior; a posterior mode requires a different limiting or neighborhood-based interpretation. Do not attach a universal optimality claim to mean, median, or mode without first declaring the loss and parameter scale.

For a binary adverse event, suppose acting costs something only when no event occurs, while waiting costs something only when it does occur. Comparing those two expected losses yields a probability threshold. The threshold is generally not one half; it depends on the costs. Costs may include resource use, delay, and effects on other people, so documenting them is part of the analysis. Sensitivity calculations reveal whether a decision survives reasonable changes in costs or depends on one debatable number.

Information can change the selected action, but its value depends on whether it can change a consequential choice. With a fixed loss function and a coherent joint model, the expected minimum posterior loss after free additional information cannot exceed the current minimum risk: ignoring the new information remains an available strategy. Actual measurement costs must then be subtracted from that expected benefit. This comparison is made before seeing the measurement, averaging over its predictive distribution; choosing only favorable possible measurements would overstate the value of information.