Statistical Decision Theory

an area of mathematical statistics and game theory that permits a number of diverse problems to be treated in a common manner. Such problems include the statistical testing of hypotheses, the obtaining of statistical estimators for parameters and of confidence limits for the estimators, and the design of experiments.

Statistical decision theory is based on the assumption that the probability distribution F of an observed random variable X_F belongs to some prior given set ℑ The principal task of statistical decision theory consists of finding the best decision function, or strategy, d = d(x) permitting a judgment to be made as to the true, but unknown, distribution F on the basis of the results of the observations x of X. In order to compare the merits of various decision functions, the loss function W[F, d(x)] is introduced. It represents the loss produced in adopting the decision d(x) (from a specified set D) when the true distribution is F.

It would be natural to regard the decision function d* = d*(x) as best if the average risk r(F, d*) = M_FW[F, d(X)] (M_F represents averaging over the distribution F) does not exceed r(F, d) for any F £ ℑ and any decision function d = d(x). In most problems, however, such a uniformly best decision function does not exist. For this reason, minimax and Bayes solutions are of great interest in statistical decision theory. The decision function d̄ = d̄(x) is said to be minimax if

The decision function d̄ = d̄(x) is said to be a Bayes solution (relative to a given prior distribution π in ℑ) if for all decision functions d

R(π, ̄) ≤ R(π, d)

where

R(π, d) = ʃ r(F,d)π(d,F)

A close relation exists between the minimax and Bayes solutions: under extremely broad assumptions regarding the data of the problem, a minimax solution is the Bayes solution relative to the least favorable prior distribution π.