  For frequentist settings in which parameter randomness represents variability
rather than uncertainty, the ideal measure of the support for one hypothesis
over another is the difference in the posterior and prior log odds. For
situations in which the prior distribution cannot be accurately estimated, that
ideal support may be replaced by another measure of support, which may be any
predictor of the ideal support that, on a per-observation basis, is
asymptotically unbiased. Two qualifying measures of support are defined. The
first is minimax optimal with respect to the population and is equivalent to a
particular Bayes factor. The second is worst-sample minimax optimal and is
equivalent to the normalized maximum likelihood. It has been extended by
likelihood weights for compatibility with more general models.
  One such model is that of two independent normal samples, the standard
setting for gene expression microarray data analysis. Applying that model to
proteomics data indicates that support computed from data for a single protein
can closely approximate the estimated difference in posterior and prior odds
that would be available with the data for 20 proteins. This suggests the
applicability of random-parameter models to other situations in which the
parameter distribution cannot be reliably estimated.
