Inequalities for predictive ratios and posterior variances in natural exponential families (Q923542)

The authors show that in the exponential family, ordering according to the predictive ratio at \(x_ 1\) and \(x_ 2\) is equivalent to ordering according to posterior variance (or covariance matrix) of the parameters, evaluated at some data point x on the line between \(x_ 1\) and \(x_ 2\). Then they set out a monotonicity property about how the posterior variance of the natural exponential family might behave as a function of the sample size hyperparameter \(n_ 0\). This monotonicity property is shown to be true for many examples: binomial, Poisson normal, multinomial, and some other cases, but does not hold uniformly in the exponential family. In fact the case of the negative binomial (multinomial) distribution is unresolved.