Abasicelementin statisticaldecisiontheory is the loss function for astatisticaldecision. In thecontextofestimation,acommonlossfunctionforanestimator of aparameter is the squared-error

Abasicelementin statisticaldecisiontheory is the loss function for astatisticaldecision.

In thecontextofestimation,acommonlossfunctionforanestimator ˆθ of aparameter θ is the squared-error loss, L(θ, ˆθ) = (ˆθ − θ)2.

The lossfunctionreferstoasinglesample,andtoevaluate ˆθ, weusetheexpectedloss, R(θ, ˆθ) = E[L(θ, ˆθ)] = ∫ L(θ, ˆθ(y))f(y; θ)dy, called the risk function. Forthesquared-errorlossfunction,thisisthemeansquarederror

(MSE). InaBayesianframework,theoverallevaluationof ˆθ is basedonthe Bayesian risk rp(ˆθ) = E[R(θ, ˆθ)] = ∫ ∫ L(θ, ˆθ(y))f(y; θ)p(θ)dydθ, whichaveragestheriskfunctionwithrespecttothepriordistribution p(θ) for θ. Anestimator that minimizestheBayesriskiscalleda Bayes estimator. ItcanbeprovedthataBayesian estimator minimizesthe posteriorexpectedloss E[L(θ, ˆθ) S y] = ∫ L(θ, ˆθ(y))g(θ S y)dθ.

Showthatforthesquared-errorlossfunction,theBayesestimatorof θ is theposteriormean,

ˆθ = E(θ S y) = ∫ θg(θ S y)dθ.