Inequalities for predictive ratios and posterior variances in natural exponential families

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 x1 and x2 is defined to be pg(x1)pg(x2)/[pg()]2 = hg(x1, x2), where pg(x1) = [integral operator] [latin small letter f with hook](x1[short parallel][theta]) g([theta]) d[theta] is the predictive distribution of x1 with respect to the prior g. We prove that hg(x1, x2) >= hg*(x1, x2) for all x1 and x2 if [latin small letter f with hook](x[short parallel][theta]) is in the natural exponential family and Covg[short parallel]x([theta]) >= Covg*[short parallel]x([theta]) in the Löwner sense, for all x on a straight line from x1 to x2. 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"