Bayesrisico

Bayesrisico, or Bayes risk, is a concept in Bayesian decision theory that measures the average loss of a decision rule when uncertainty about an unknown parameter is described by a prior distribution. It combines the loss function, the data-generating process, and the prior to assess overall performance across possible parameter values.

Mathematically, for a decision rule δ that maps observations x to actions a = δ(x) and a loss function L(a, θ) for parameter θ, the Bayes risk is defined as R_B(δ) = ∫∫ L(δ(x), θ) p(x|θ) π(θ) dθ dx. Equivalently, R_B(δ) = ∫ r(x, δ(x)) m(x) dx, where r(x, a) = ∫ L(a, θ) π(θ) p(x|θ) dθ is the conditional risk and m(x) = ∫ p(x|θ) π(θ) dθ is the marginal distribution of x under the prior. The Bayes rule δ* minimizes this Bayes risk.

For a fixed x, the posterior expected loss ρ(a|x) = E[L(a, θ) | X = x] is minimized by the Bayes decision δ*(x). Common losses give familiar rules: with squared error loss L(a, θ) = (a − θ)^2, the Bayes estimator is the posterior mean; with 0–1 loss, the Bayes estimator is the posterior mode (MAP).

Bayes risk contrasts with frequentist risk and minimax risk, which do not average over a prior. In practice, Bayes risk informs the choice of decision rules and priors by evaluating average performance under the assumed prior distribution.