5.2 Bayes Rule and Risk

Bayesian rule and risk

The expected loss of a decision $d\in\mathcal{D}$ with respect to the state of nature $\theta\in\Theta$ is

	$\displaystyle\rho(\pi,d)$	$\displaystyle=$	$\displaystyle\mbox{\rm E\,}_{\pi}L(d,\theta)$
		$\displaystyle=$	$\displaystyle\int_{\theta}L(d,\theta)\pi(\theta)d\theta$

Bayes risk

$\rho^{\star}(\pi)$ is the minimum expected loss (with respect to $\pi(\theta)$ )

\displaystyle\rho^{\star}(\pi)=\underset{d\in\mathcal{D}}{\inf}\;\;\rho(\pi,d)

(5.2)

Bayes rule

is the decision, $d^{\star}$ , that minimises Equation 5.2.