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In the typical analysis of a data set, a single method is selected forstatistical reporting even when equally applicable methods yield very differentresults. Examples of equally applicable methods can correspond to those ofdifferent ancillary statistics in frequentist inference and of different priordistributions in Bayesian inference. More broadly, choices are made betweenparametric and nonparametric methods and between frequentist and Bayesianmethods.  Rather than choosing a single method, it can be safer, in a game-theoreticsense, to combine those that are equally appropriate in light of the availableinformation. Since methods of combining subjectively assessed probabilitydistributions are not objective enough for that purpose, this paper introducesa method of distribution combination that does not require any assignment ofdistribution weights. It does so by formalizing a hedging strategy in terms ofa game between three players: nature, a statistician combining distributions,and a statistician refusing to combine distributions. The optimal move of thefirst statistician reduces to the solution of a simpler problem of selecting anestimating distribution that minimizes the Kullback-Leibler loss maximized overthe plausible distributions to be combined. The resulting combined distributionis a linear combination of the most extreme of the distributions to be combinedthat are scientifically plausible. The optimal weights are close enough to eachother that no extreme distribution dominates the others.  The new methodology is illustrated by combining conflicting empirical Bayesmethodologies in the context of gene expression data analysis.

Resolving conflicts between statistical methods by probability combination: Application to empirical Bayes analyses of genomic data