The Minimal Belief Principle: a New Method for Parametric Inference

Contemporary very-high-dimensional (VHD) statistical problems call attention more than ever to solving the fundamental problem of scientific inference, that is, to make situation-specific inference with credible evidential support. After scrutinizing the great innovative ideas behind Fisher’s fiducial argument and the Dempster-Shafer (DS) theory for scientific inference, we recognize that given a postulated sampling model, reasoning for statistical inference (about a particular realization of random variables) should be different from reasoning for data generation. The classical belief in distributional invariance of pivotal variables does not distinguish these two types of reasoning processes and is thus often too strong to be believable. Intuitively, beliefs with higher credibility can be obtained from the classical belief by making it weaker. This general idea is termed as the “minimal belief” (MB) principle. Technically, the proposed method is built on the DS theory, and provides ways to capture realistically more “don’t know” and thereby to build better DS models for solving VHD problems. It is shown that for general single-parameter and certain multiparameter distributions, the MB posteriors are obtained in closed form. The method is illustrated with a variety of examples, including the simple test of significance, the Behrens-Fisher problem, the multinomial model, and the many-normal-means problem. The many-normal-means example offers an MB perspective of often-crude Bayesian and related shrinkage techniques, which have been considered necessary in the last half a century.