Bayesian Modelling and Sensitivity Analysis

Abstract

Numerical integration techniques now exist which permit a very flexible approach to Bayesian modelling. Low dimensional summaries can be extracted from joint posterior distributions of up to 15 dimensions. The sensitivity of particular summaries to changes in both the model and the prior can thus be investigated. This paper discusses the various aspects of sensitivity in a Bayesian analysis and demonstrates something of the power of numerical integration via two examples. Practical data analysis consistent with the Bayesian paradigm has recently been given a substantial boost with the development of efficient high-dimensional numerical integration methods. These allow the computation of posterior moments and low dimensional summaries such as univariate and bivariate marginal distributions from high-dimensional posterior densities. For early examples using Monte Carlo integra- tion methods, see Stewart (1979) or van Dijk & Kloek (1978). Naylor & Smith (1982) describe an iterative procedure which makes repeated use of Gauss-Hermite rules over three dimensional cartesian product grids. A general review of progress in this area is given by Smith et al. (1985). In 1983, a research project funded by the SERC was established at the University of Nottingham to extend the ideas of Naylor & Smith to higher dimensions. That project has confirmed that numerical integration in up to 15 dimensions can be carried out routinely, if mixed strategies involving cartesian product rules, spherical rules and Monte Carlo procedures are used. Details are given by Shaw (1985a,b) and will be published elsewhere. Freed from constraints imposed by analytical tractability, a Bayesian analysis is straightforward and extremely flexible. In computing terms, all that is required is a routine for evaluating the likelihood for selected values of the model parameters. Prior distributions need not be restricted to conjugate classes and can more effectively represent prior opinion. The sensitivity of particular inferences to changes in the form of either the model or the prior can be readily investigated. This paper presents two examples of statistical modelling in a Bayesian framework, to demonstrate both the scope of current integration methods and the ease with which a sensitivity analysis can be carried out. Although a Bayesian analysis is commonly portrayed as the revision of subjective beliefs in the light of available data, the idea of a Bayesian sensitivity analysis is not new. It has frequently been advocated that a single data set should motivate many prior to posterior analyses. These arguments are reviewed in Section 2. What is still needed is a practical definition of sensitivity in this context. Consider a graph in which several versions of a marginal posterior density are superimposed. The fact that some of these curves are distinguishable from each other, at the level of resolution employed by the display, does not automatically imply that the margin is sensitive to choice of either the model or the prior. In some circum- stances, gross discrepancies in the tails of the distributions may be unimportant providing the mode or the mean is stable. In other circumstances, the estimation of extreme tail area probabilities may be the sole purpose of the analysis. One possibility