Bayesian Analysis, Model Selection and Prediction

Summary

Introduction

In this paper, we shall consider (1) the derivation of prior distributions for use in Bayesian analysis, (2) Bayesian and non-Bayesian model selection procedures and (3) an application of Bayesian model selection procedures in the context of an applied forecasting problem. With respect to derivations of prior distributions, Jaynes (1980, p.618) has stated:

“It was from studying these frashes of intuition in Jeffreys that I became convinced that there must exist a general formal theory of determination of priors by logical analysis of the prior information–and that to develop it is today the top priority research problem of Bayesian theory.”

As is well known, many including Jeffreys (1967), Lindley (1956), Savage (1961), Hartigan (1964), Novick and Hall (1965), Jaynes (1968), Box and Tiao (1973), Bernardo (1979), and Zellner (1971, 1977, 1990) have worked to solve the “top priority research problem of Bayesian theory,” to use Jaynes’ words. Herein we shall consider further the maximal data information prior (MDIP) distribution approach put forward and discussed in Zellner (1971, 1977, 1990). A MDIP is a solution to a well-defined optimization problem and its use provides maximal data information relative to the information in a prior density. Also, as shown below, a MDIP maximizes the expectation of the logarithm of the ratio of the likelihood function, ℓ(θ|y), to the prior density, π(θ), i.e. ℓn[ℓ(θ|y)/π(θ)]. Several explicit MDIPs will be presented including those for parameters of time series processes–see Phillips (1991) for a lively discussion of this topic.