User:Slava3087

In classical statistical decision theory, where we are faced with the problem of estimating a parameter (vector) $x$ from observations $y$ with respect to a loss function

we have an estimator $\delta$ that is used to estimate a parameter $\theta \in \Theta$. We also assume a risk function $R(\theta ,\delta )$, usually specified as the integral of a loss function. In this framework, ${\tilde {\delta }}$ is called minimax if it satisfies

$\sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\sup _{\theta }R(\theta ,\delta )$.

An alternative criterion in the decision theoretic framework is the Bayes estimator in the presence of a prior distribution $\Pi$. An estimator is Bayes if it minimizes the average risk

$\int _{\Theta }R(\theta ,\delta )\,d\Pi (\theta )$