Comparison of Experiments and Information Measures

Abstract

Let Ex = {X, SX; Pθ, θ∈Θ} and EY = {Y, SY; Qθ, θ∈Θ} be two statistical experiments with the same parameter space Θ. Some implications of the sufficiency of EX for EY, according to Blackwell's definition, are given in terms of Kullback-Leibler information and Fisher information matrices. For a scale parameter θ, and $k_1 k_2 0$, the experiment with parameter θk1 is proved to be sufficient for the experiment with parameter θk2 for a class of distributions including the gamma distribution and the normal distribution with known mean. Some results of Stone are generalized to the class of experiments with both location and scale parameters. A concept of sufficiency is proposed in which EX is more informative than EY for a fixed prior distribution of θ if the expected Bayes risk from EX is not greater than that from EY for every decision problem involving θ. This concept is then used to develop a definition of marginal Bayesian sufficiency in the presence of nuisance parameters.