31.1 Outline of decision analysis
Following Andrew Gelman et al. (2013), Bayesian decision analysis can be factored into the following four steps.
Define a set \(X\) of possible outcomes and a set \(D\) of possible decisions.
Define a probability distribution of outcomes conditional on decisions through a conditional density function \(p(x \mid d)\) for \(x \in X\) and \(d \in D.\)
Define a utility function \(U : X \rightarrow \mathbb{R}\) mapping outcomes to their utility.
Choose action \(d^* \in D\) with highest expected utility, \[ d^* = \textrm{arg max}_d \ \mathbb{E}[U(x) \mid d]. \]
The outcomes should represent as much information as possible that is relevant to utility. In Bayesian decision analysis, the distribution of outcomes will typically be a posterior predictive distribution conditioned on observed data. There is a large literature in psychology and economics related to defining utility functions. For example, the utility of money is usually assumed to be strictly concave rather than linear (i.e., the marginal utility of getting another unit of money decreases the more money one has).