Chapter 5 Week 5 Bayesian statistics: Decisions

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  Ingredients of a Decision problem

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  Bayes rule and Bayes risk

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  Point estimation

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  Bayes rule and Bayes risk for an experiment

The notes so far have described methods for conducting statistical inference. Estimation, prediction and model selection have all been carried out using probability as the sole measure of uncertainty. In this chapter we add the other ingredient of the Bayesian system and that is decisions or actions. The usefulness of decisions to the decision-maker is subjective and measured by a utility function. The concept of utility enables us to use estimates and judgments that reflect the point of the analysis. Utilities can also be expressed in terms of rewards such as profit. Bayesian decision making involves selecting decisions by using current beliefs or knowledge (expressed as probabilities) to maximize the expected utility of the decision maker.

We use $\pi (\theta \in \mathrm{\Theta })$ to depict our belief of the state of nature. The utility of a decision $d\in 𝒟$ is expressed as $U(d,\theta )$, the utility of choosing $d$ when $\theta$ is the true value. We shall define loss to be

This loss or a utility can measure the decision-maker’s perception of a reward, $L(r(\theta ),d)$ where $r\in ℛ$.