Abstract-In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estima-tion, universal data compression, composite hypothesis testing, and stock-market portfolio selection. 1.

In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P-values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications. The points we emphasize are:- from Jeffreys's Bayesian point of view, the purpose of hypothesis testing is to evaluate the evidence in favor of a scientific theory;- Bayes factors offer a way of evaluating evidence in favor ofa null hypothesis;- Bayes factors provide a way of incorporating external information into the evaluation of evidence about a hypothesis;- Bayes factors are very general, and do not require alternative models to be nested;- several techniques are available for computing Bayes factors, including asymptotic approximations which are easy to compute using the output from standard packages that maximize likelihoods;- in "non-standard " statistical models that do not satisfy common regularity conditions, it can be technically simpler to calculate Bayes factors than to derive non-Bayesian significance

The Schwarz (1978) information criterion, SIC, is a widely-used tool in model selection, largely due to its computational simplicity and effective performance in many modeling frameworks. The derivation of SIC (Schwarz, 1978) establishes the criterion as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. In this paper, we investigate the derivation for the identification of terms which are discarded as being asymptotically negligible, but which may be significant in small to moderate sample-size applications. We suggest several SIC variants based on the inclusion of these terms. The results of a simulation study show that the variants improve upon the performance of SIC in two important areas of application: multiple linear regression and time series analysis. 1. Introduction One of the most important problems confronting an investigator in statistical modeling is the choice of an appropriate model to characterize the underlyin...

Laplace's method, a family of asymptotic methods used to approximate integrals, is presented as a potential candidate for the tool box of techniques used for knowledge acquisition and probabilistic inference in belief networks with continuous variables. This technique approximates posterior moments and marginal posterior distributions with reasonable accuracy [errors are O(n,2) for posterior means] in many interesting cases. The method also seems promising for computing approximations for Bayes factors for use in the context of model selection, model uncertainty and mixtures of pdfs. The limitations, regularity conditions and computational di culties for the implementation of Laplace's method are comparable to those associated with the methods of maximum likelihood and posterior mode analysis. 1

The Schwarz information criterion (SIC, BIC, SBC) is one of the most widely known and used tools in statistical model selection. The criterion was derived by Schwarz (1978) to serve as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. Although the original derivation assumes that the observed data is independent, identically distributed, and arising from a probability distribution in the regular exponential family, SIC has traditionally been used in a much larger scope of model selection problems. To better justify the widespread applicability of SIC, we derive the criterion in a very general framework: one which does not assume any specific form for the likelihood function, but only requires that it satisfies certain non-restrictive regularity conditions.

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