General Theory of Inferential Models I. Conditional Inference

As applied problems have grown more complex, statisticians have been gradually led to reconsider the foundations of statistical inference. The recently proposed inferential model (IM) framework of Martin, Zhang and Liu (2010) achieves an interesting compromise between the Bayesian and frequentist ideals. Indeed, inference is based on posterior probability-like quantities, but there are no priors and the inferential output satisfies certain desirable long-run frequency properties. In this two-part series, we further develop the theory of IMs into a general framework for statistical inference. Here, in Part I, we build on the idea of making inference by predicting unobserved auxiliary variables, focusing primarily on an intermediate step of conditioning, whereby the dimension of this auxiliary variable is reduced to a manageable size. This dimension reduction step leads to a simpler construction of IMs having the required longrun frequency properties. We show that under suitable conditions, this dimension reduction step can be made without loss of information, and that these conditions are satisfied in a wide class of models, including those with a group invariance structure. The important credibility theorem of Zhang and Liu (2010) is extended to handle the case of conditional IMs, and connections to conditional inference in the likelihood framework are made which, in turn, allow for numerical approximation of the conditional posterior belief functions using the well-known “magic formula” of Barndorff-Nielsen (1983). The conditional IM approach is illustrated on a variety of examples, including Fisher’s problem of the Nile.