Bayeisan decision theory¶

It is a way to connect our belief into actions that are optimal. We can formalize this problem as a game against nature. In this game nature pics a state \(y \in Y\), unknow to us, and then generates an observation \(x \in X\),which we get to see. Now we have to make a decision to choose an action a from some action space \(\mathcal{A}\). Then we measure our loss \(L(y,a)\), which measures how compatible our cation a is with natures hidden state y.

Our goal is to minimize this loss over all states, this is done by minimizng the expected loss:

\[ \delta(x) = \arg_{a \in \mathcal{A}} \min E[L(y,a)]\]

Instead of loss function we can define a utility function

\[U(y,a) = -L(y,a)\]

Than we want to maximize the expected utility:

\[ \delta(x) = \arg_{a \in \mathcal{A}} \max E[U(y,a)] \]

This is also called the maximum utlity principle, or simply rational behaviour

In a bayesian setting if we talk about expected loss, we mean posterior expected loss

\[p(a|x) \triangleq E_{p(y|x)}[L(y,a)] = \sum_x L(y,a)p(y|x) \]

Hence the Bayesian estimator is:

\[\delta(x) = \arg_{a \in \mathcal{A}} \min p(a|x)\]

In classificatin setings we may care about false positive false negative tradeoff

Dirichlet Multinomial model Steps of HMC iteration