Combining Statistical Evidence When Evidence Is Measured by Relative Belief

The problem of combining statistical evidence concerning an unknown, contained in each of the k Bayesian inference bases, is discussed. This can be considered as being related to the problem of pooling k priors to determine a consensus prior, but the focus here is instead on combining a measure of statistical evidence to obtain a consensus measure of statistical evidence. The linear opinion pool is seen to have the most appropriate properties for this role. In particular, linear pooling preserves a consensus with respect to the evidence, and other rules do not. While linear pooling does not preserve prior independence, it is shown that it still behaves appropriately with respect to the expression of statistical evidence in such a context. For the more general problem of combining statistical evidence, where the priors as well as the sampling models may differ, Jeffrey conditionalization plays a key role.