Bayes optimal decision rule

The aim is to find an optimal decision rule to choose between competing hypotheses. If the prior probabilities are fixed

$$\textrm{Decide } H_i \textrm{ if } P(H_i)>P(H_j) \forall i \neq j$$

The optimal decision rule gives the minimum error rate possible if we are not allowed to observe the pattern. Can we make a better decision if more information is available?

Remember Bayes rule

$$\begin{array}{lll} P(H_i\vert Z) & = & \frac{P(Z\vert H_i)... ...P(H_i) } \textrm{ (through marginilisation) }\\ \end{array}$$

where $P(H_i\vert Z)$ denotes the posterior probability, $P(Z\vert H_i)$ the likelihood, $P(H_i)$ the prior probability and $P(Z)$ the evidence.

Bayes decision rule (with more information)

$$\begin{array}{llll} \textrm{Decide } H_i \textrm{ if } & P... ...)}{P(Z\vert H_j)} & > & \frac{P(H_j)}{P(H_i)}\\ \end{array}$$