A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

ABSTRACT

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi to a well-known identity satisfied by the Bayes factor. Supplementary materials for this article are available online.     

Bayes Factor Consistency for One-way Random Effects Model

In this article, we consider Bayesian hypothesis testing for the balanced one-way random effects model. A special choice of the prior formulation for the ratio of variance components is shown to yield an explicit closed-form Bayes factor without integral representation. Furthermore, we study the consistency issue of the resulting Bayes factor under three asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. Finally, the behavior of the proposed approach is illustrated by simulation studies.     

Bayes factor consistency for unbalanced ANOVA models

In practical situations, most experimental designs often yield unbalanced data which have different numbers of observations per unit because of cost constraints, missing data, etc. In this paper, we consider the Bayesian approach to hypothesis testing or model selection under the one-way unbalanced fixed-effects analysis-of-variance (ANOVA) model. We adopt Zellner's g-prior with the beta-prime distribution for g, which results in an explicit closed-form expression of the Bayes factor without integral representation. Furthermore, we investigate the model selection consistency of the Bayes factor under three different asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. The results presented extend some existing ones of the Bayes factor for the balanced ANOVA models in the literature.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Properties of Bayes Factors Based on Test Statistics

Abstract.  This article examines the consistency, interpretation and application of Bayes factors constructed from standard test statistics. Primary conclusions are that Bayes factors based on multinomial and normal test statistics are consistent for suitable choices of the hyperparameters used to specify alternative hypotheses, and that such constructions can be extended to obtain consistent Bayes factors based on likelihood ratio statistics. A connection between Bayes factors based on likelihood ratio statistics and the Bayesian information criterion is exposed, as is a connection between Bayes factors based on F statistics and parametric Bayes factors based on normal-inverse gamma models. Similarly, Bayes factors based on chi-squared statistics for multinomial data are shown to provide accurate approximations to Bayes factors based on multinomial/Dirichlet models. An illustration of how the simple form of these Bayes factors can be exploited to generate easily interpretable summaries of the experimental 'weight of evidence' is provided.     

A ROBUST BAYES FACTOR FOR LINEAR MODELS

This paper proposes a new robust Bayes factor for comparing two linear models. The factor is based on a pseudo‐model for outliers and is more robust to outliers than the Bayes factor based on the variance‐inflation model for outliers. If an observation is considered an outlier for both models this new robust Bayes factor equals the Bayes factor calculated after removing the outlier. If an observation is considered an outlier for one model but not the other then this new robust Bayes factor equals the Bayes factor calculated without the observation, but a penalty is applied to the model considering the observation as an outlier. For moderate outliers where the variance‐inflation model is suitable, the two Bayes factors are similar. The new Bayes factor uses a single robustness parameter to describe a priori belief in the likelihood of outliers. Real and synthetic data illustrate the properties of the new robust Bayes factor and highlight the inferior properties of Bayes factors based on the variance‐inflation model for outliers.     

On asymptotic properties of Bayesian partially linear models

In this paper, we present large sample properties of a partially linear model from the Bayesian perspective, in which responses are explained by the semiparametric regression model with the additive form of the linear component and the nonparametric component. For this purpose, we investigate asymptotic behaviors of posterior distributions in terms of consistency. Specifically, we deal with a specific Bayesian partially linear regression model with additive noises in which the nonparametric component is modeled using Gaussian process priors. Under the Bayesian partially linear model using Gaussian process priors, we focus on consistency of posterior distribution and consistency of the Bayes factor, and extend these results to generalized additive regression models and study their asymptotic properties. In addition we illustrate the asymptotic properties based on empirical analysis through simulation studies.     

Bayes factors for goodness of fit testing

We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a non-asymptotic method that can be used to quantify the evidence for or against a sub-model. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor.     

Nonparametric bayes estimation of the survival function from a record of failures and follow-ups

In some observational studies, we have random censoring model. However, the data available may be partially observable censored data consisting of the observed failure times and only those nonfailure times which are subject to follow-up. Suzuki (1985) discussed the problem of nonparametric estimation of the survival function from such partially observable censored data. In this article, we derive a nonparametric Bayes estimator of the survival function for such data of failures and follow-ups under a Dirichlet process prior and squared error loss. The limiting properties such as the mean square consistency, weak convergence and strong consistency of the Bayes estimator are studied. Finally, the procedures developed are illustrated by means of an example.     

Computing some bayes estimates for the mean of a normal distribution

For sampling from a normal population with unknown mean, two families of prior densities for the mean are discussed. The corresponding posterior densities are found. A data analyst may choose a prior from these families to represent prior beliefs and then compute the corresponding Bayes estimator, using the techniques discussed.     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Alternative Procedures to Discriminate Non Nested Multivariate Linear Regression Models

ABSTRACT

This article builds classical and Bayesian testing procedures for choosing between non nested multivariate regression models. Although there are several classical tests for discriminating univariate regressions, only the Cox test is able to consistently handle the multivariate case. We then derive the limiting distribution of the Cox statistic in such a context, correcting an earlier derivation in the literature. Further, we show how to build alternative Bayes factors for the testing of nonnested multivariate linear regression models. In particular, we compute expressions for the posterior Bayes factor, the fractional Bayes factor, and the intrinsic Bayes factor.     

Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors

In objective Bayesian model selection, a well-known problem is that standard non-informative prior distributions cannot be used to obtain a sensible outcome of the Bayes factor because these priors are improper. The use of a small part of the data, i.e., a training sample, to obtain a proper posterior prior distribution has become a popular method to resolve this issue and seems to result in reasonable outcomes of default Bayes factors, such as the intrinsic Bayes factor or a Bayes factor based on the empirical expected-posterior prior.     

Properties of bagged nearest neighbour classifiers

Summary.  It is shown that bagging, a computationally intensive method, asymptotically improves the performance of nearest neighbour classifiers provided that the resample size is less than 69% of the actual sample size, in the case of with-replacement bagging, or less than 50% of the sample size, for without-replacement bagging. However, for larger sampling fractions there is no asymptotic difference between the risk of the regular nearest neighbour classifier and its bagged version. In particular, neither achieves the large sample performance of the Bayes classifier. In contrast, when the sampling fractions converge to 0, but the resample sizes diverge to ∞, the bagged classifier converges to the optimal Bayes rule and its risk converges to the risk of the latter. These results are most readily seen when the two populations have well-defined densities, but they may also be derived in other cases, where densities exist in only a relative sense. Cross-validation can be used effectively to choose the sampling fraction. Numerical calculation is used to illustrate these theoretical properties.     

Admissibility of estimators in the non-regular family under entropy loss function

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities. The admissibility of generalized Bayes and Pitman estimates of functions of parameters have been established under entropy loss function.     

Admissibility of generalized bayes and pitman estimates in the non-regular family

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities Following Karlin s (1958) technique, we have established the ad-missibility of generalized Bayes estimates and Pitman estimates. Some examples are discussed.     

Calibrated Bayes factors for model comparison

A Bayes factor between two models can be greatly affected by the prior distributions on the model parameters. When prior information is weak, very dispersed proper prior distributions are known to create a problem for the Bayes factor when competing models differ in dimension, and it is of even greater concern when one of the models is of infinite dimension. Therefore, we propose an innovative method which uses training samples to calibrate the prior distributions so that they achieve a reasonable level of ‘information’. Then the calibrated Bayes factor can be computed over the remaining data. This method makes no assumption on model forms (parametric or nonparametric) and can be used with both proper and improper priors. We illustrate, through simulation studies and a real data example, that the calibrated Bayes factor yields robust and reliable model preferences under various situations.     

INTRINSIC PRIORS FOR TWO-SAMPLE TESTS IN NORMAL POPULATIONS

ABSTRACT

There have been considerable amounts of work regarding the development of various default Bayes factors in model selection and hypothesis testing. Two commonly used criteria, the intrinsic Bayes factor and the fractional Bayes factor are compared to test two independent normal means and variances. We also derive several intrinsic priors whose Bayes factors are asymptotically equivalent to the respective Bayes factors. We demonstrate our results in simulated datasets.     

Empitical bayes estimations for some distribution families

Some estimates of prior density based on orthogonal expansions are proposed for some family of conditional densities. Their related properties are studied. The associated empirical Bayes estimators are also proposed. Three examples are illustrated and some of its Monte Carlo results are also given.