TESTING A PRECISE HYPOTHESIS INTERPRETING P-VALUES FROM A ROBUST BAYESIAN VIEW POINT (LOWER BOUNDS, POSTERIOR PROBABILITIES)

MOHAN DELAMPADY, Purdue University

Abstract

The testing of two-sided hypotheses in univariate and multivariate situations is considered. The goal is to establish lower bounds on the posterior probabilities of null hypotheses, using Robust Bayesian techniques, and to compare these lower bounds with the corresponding P-values. The lower bounds are calculated over classes of prior distributions which (1) assign specified probabilities to each hypotheses; and (2) can otherwise be considered to be "objective". An example would be the class of all prior distributions which assign probability 1/2 to each hypothesis and are suitably "symmetric". Dealing with such classes of priors imbues the lower bounds on posterior probability with an objectivity that is lacking in Bayesian analyses with a specified prior. This objectivity is also indicated by the fact that the results can be interpreted as bounds on the likelihood ratio of the hypotheses, if one were to take a likelihood approach to testing. The interest in these lower bounds, besides their intrinsic Bayesian interest, is that they tend to be considerably larger than the corresponding P-values. While it is well understood that P-values and posterior probabilities are very different quantities the magnitudes of the differences that we observe make clear the need for very careful interpretation of P-values.

Degree

Ph.D.

Subject Area

Statistics

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