Reasoning to a Foregone Conclusion

When can a Bayesian select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be greater than its prior probability? We discuss an elementary result that establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. We illustrate how when the sufficient conditions fail, because probability is finitely but not countably additive, it may be that a Bayesian can design an experiment to lead his/her posterior probability into a foregone conclusion. The problem has a decision theoretic version in which a Bayesian might rationally pay not to see the outcome of certain cost-free experiments, which we discuss from several perspectives. Also, we relate this issue in Bayesian hypothesis testing to various concerns about "optional stopping."