Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio

Abstract We present a simple method for computing Bayes factors. The method derives from observing that in general, a Bayes factor can be written as the product of a quantity called the Savage-Dickey density ratio and a correction factor; both terms are easily estimated from posterior simulation. In some cases it is possible to do these computations without ever evaluating the likelihood.

[1]  H. Jeffreys,et al.  Theory of probability , 1896 .

[2]  J. Taylor,et al.  Censored Observations in Randomized Block Experiments , 1959 .

[3]  J. Dickey,et al.  The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain , 1970 .

[4]  J. Dickey.  The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters , 1971 .

[5]  J. Dickey,et al.  Bayes factors for independence in contingency tables , 1974 .

[6]  J. Dickey.  Approximate Posterior Distributions , 1976 .

[7]  Peter E. Rossi,et al.  Bayesian analysis of dichotomous quantal response models , 1984 .

[8]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[9]  J. Geweke,et al.  Bayesian Inference in Econometric Models Using Monte Carlo Integration , 1989 .

[10]  L. Tierney,et al.  Fully Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions , 1989 .

[11]  L. Joseph,et al.  Bayesian Statistics: An Introduction , 1989 .

[12]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[13]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[14]  R. Kass,et al.  Approximate Bayes Factors and Orthogonal Parameters, with Application to Testing Equality of Two Binomial Proportions , 1992 .

[15]  Peter E. Rossi,et al.  Bayes factors for nonlinear hypotheses and likelihood distributions , 1992 .

[16]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[17]  L. Tierney.  Exploring Posterior Distributions Using Markov Chains , 1992 .

[18]  M. Newton.  Approximate Bayesian-inference With the Weighted Likelihood Bootstrap , 1994 .

[19]  Ming-Hui Chen.  Importance-Weighted Marginal Bayesian Posterior Density Estimation , 1994 .

[20]  A. Gelfand,et al.  Bayesian Model Choice: Asymptotics and Exact Calculations , 1994 .

[21]  S. Chib.  Marginal Likelihood from the Gibbs Output , 1995 .

[22]  Xiao-Li Meng,et al.  SIMULATING RATIOS OF NORMALIZING CONSTANTS VIA A SIMPLE IDENTITY: A THEORETICAL EXPLORATION , 1996 .