The posterior risk 
of an action 
after observing data 
is the expected value of its loss
function with respect to the posterior
distribution. If 
is a loss function and 
is a density for the posterior
distribution of the unknown parameter 
, then
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For a discrete parameter, the integral is replaced by a sum. A Bayes
decision rule selects, for each observed 
, an action that minimizes the posterior risk.
