A general Bayesian model for hierarchical inference

Abstract

An inductive inference problem will often be structured so that the target variable (hypotheses) are logically distant from the observable events (data). In this situation it may be difficult or impossible to assess the probabilistic connection between them, but it may be possible to decompose the problem through the use of intermediate or explanatory variables. That is, it will often be possible to assess the likelihood of the observed data given some intermediate variable, and the likelihood of that intermediate variable given another, and so on, until the hypotheses of interest are reached. Inferences which incorporate one or more intermediate variables are called hierarchical, cascaded, or multistage inferences. The present paper presents a normative model for the solution of the general hierarchical inference problem. The formulation begins with a formal description of the hierarchical inference tree, including a discussion of various simplifying conditional independence assumptions. The solution is first derived for three special case models of differing structure, and then the algorithm for the general solution is given for two cases: one in which the conditional independence assumptions have been made and one in which they have not.