REASONING TO A FORGONE CONCLUSION

When can a Bayesian select an hypothesis H and design an experiment (o r a sequence of experiments) to make certain that, given the experimen tal outcome(s), the posterior probability of H will be greater than it s prior probability? We discuss an elementary result that establishes sufficient conditions under which this reasoning to a foregone conclus ion cannot occur. We illustrate how when the sufficient conditions fai l, 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 the oretic version in which a Bayesian might rationally pay not to see the outcome of certain cost-free experiments, which we discuss from sever al perspectives. Also, we relate this issue in Bayesian hypothesis tes ting to various concerns about ''optional stopping.''