Statistical decisions are decisions made on the basis of observations of a phenomenon that obeys probabilistic laws that are not completely known (seePROBABILITY) .
As an example, let us consider the disinfection of water by chlorination. The amount of chlorine to be added should depend on the average number θ of bacteria per unit volume. The value of θ, however, is not known and is estimated from the results X1, X2,....Xn of a computation of the number of bacteria in n independently selected unit volumes of water. In the simplest model it is assumed that Xi, for i = 1,...,n, has a Poisson distribution with the unknown mean (mathematical expectation) θ. The statistical decision as to the amount of chlorine to be added will therefore be a function of a statistical estimator θ* of the parameter θ. In selecting θ* there must be taken into account the undesirable consequences of both an underestimate of θ (insufficient disinfection of the water) and an overestimate of θ (worsening of the taste of the water owing to excessive chlorination).
Statistical decision theory provides a precise mathematical formulation of the concepts pertaining to statistical decisions and to methods of comparing statistical decisions.
IU. V. PROKHOROV