Models and Selection Criteria for Regression and Classification

TL;DR

This paper explores Bayesian models for regression and classification, emphasizing the importance of prior knowledge, and compares two Bayesian model selection criteria within a unified prequential framework.

Contribution

It introduces the analysis of Bayesian regression/classification models, critiques the practice of transforming models to BRC form, and compares two Bayesian model selection criteria.

Findings

Transforming arbitrary models to BRC can ignore prior knowledge.

Two Bayesian model selection criteria are contrasted and conditions for their agreement are provided.

The prequential framework helps evaluate model selection methods.

Abstract

When performing regression or classification, we are interested in the conditional probability distribution for an outcome or class variable Y given a set of explanatoryor input variables X. We consider Bayesian models for this task. In particular, we examine a special class of models, which we call Bayesian regression/classification (BRC) models, that can be factored into independent conditional (y|x) and input (x) models. These models are convenient, because the conditional model (the portion of the full model that we care about) can be analyzed by itself. We examine the practice of transforming arbitrary Bayesian models to BRC models, and argue that this practice is often inappropriate because it ignores prior knowledge that may be important for learning. In addition, we examine Bayesian methods for learning models from data. We discuss two criteria for Bayesian model selection that are appropriate for repression/classification: one described by Spiegelhalter et al. (1993), and another by Buntine (1993). We contrast these two criteria using the prequential framework of Dawid (1984), and give sufficient conditions under which the criteria agree.