ABSTRACT

When can a Bayesian investigator 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 lower than its prior probability? We report an elementary result which establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. Through an example, we discuss how this result extends to the perspective of an onlooker who agrees with the investigator about the statistical model for the data but who holds a different prior probability for the statistical parameters of that model. We consider, specifically, one-sided and two-sided statistical hypotheses involving i.i.d. Normal data with conjugate priors. In a concluding section, using an “improper” prior, we illustrate how the preceding results depend upon the assumption that probability is countably additive.

EXPECTED CONDITIONAL PROBABILITIES AND REASONING TO FOREGONE CONCLUSIONS

Suppose that an investigator has his or her designs on rejecting, or at least making doubtful, a particular statistical hypothesis H. To what extent does basic inductive methodology insure that, without violating the total evidence requirement, this intent cannot be sure to succeed? We distinguish two forms of the question:

1. Can the investigator plan an experiment so that he or she is certain it will halt with evidence that disconfirms W.

2. Can the investigator plan an experiment so that others are certain that the investigator will halt with evidence that disconfirms H?