Comparison of Experiments and Information Measures

Abstract

Let $\mathscr{E}_x = \{X, S_X; P_\theta, \theta\in\Theta\}$ and $\mathscr{E}_Y = \{Y, S_Y; Q_\theta, \theta\in\Theta\}$ be two statistical experiments with the same parameter space $\Theta$. Some implications of the sufficiency of $\mathscr{E}_X$ for $\mathscr{E}_Y$, according to Blackwell's definition, are given in terms of Kullback-Leibler information and Fisher information matrices. For a scale parameter $\theta$, and $k_1 > k_2 > 0$, the experiment with parameter $\theta^{k_1}$ is proved to be sufficient for the experiment with parameter $\theta^{k_2}$ 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 $\mathscr{E}_X$ is more informative than $\mathscr{E}_Y$ for a fixed prior distribution of $\theta$ if the expected Bayes risk from $\mathscr{E}_X$ is not greater than that from $\mathscr{E}_Y$ for every decision problem involving $\theta$. This concept is then used to develop a definition of marginal Bayesian sufficiency in the presence of nuisance parameters.