Inequalities for predictive ratios and posterior variances in natural exponential families

Abstract

The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x 1 and x 2 is defined to be p g ( x 1) p g ( x 2)/[ p g ( x )] 2 = h g ( x 1, x 2), where p g ( x 1) = ∫ ƒ( x 1∥θ) g(θ) dθ is the predictive distribution of x 1 with respect to the prior g. We prove that h g ( x 1, x 2) ≥ h g ∗( x 1, x 2) for all x 1 and x 2 if ƒ( x∥θ) is in the natural exponential family and Cov g∥ x (θ) ≥ Cov g ∗∥ x(θ) in the Loewner sense, for all x on a straight line from x 1 to x 2. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g ∗ differ in that the “sample size”