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Bayes Risk

In a Bayesian approach, the expectation is calculated using the prior distribution π of the parameter θ:

where m(x) is known as the predictive likelihood wherein θ has been "integrated out," π (θ | x) is the posterior distribution, and the order of integration has been changed. One then should choose the action a which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside dx is known as the Posterior Risk, and minimising it with respect to decision a also minimizes the overall Bayes Risk. This optimal decision, a is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ, over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ.