Fractional Bayes Factor Research Articles

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.

On the asymptotic stability of the intrinsic and fractional bayes factors for testing some diffusion models

Random processes, from which a single sample path data are available on a fine time scale, abound in many areas including finance and genetics. An effective way to model such data is to consider a suitable continuous-time-scale analog, X t say, for the underlying process. We consider three diffusion models for the process X t and address model selection under improper priors. Specifically, fractional and intrinsic Bayes factors (FBF and IBF) for model selection are considered. Here, we focus on the asymptotic stability of the IBF's and FBF's for comparing these models. Specifically, we propose to employ certain novel transformations of the data in order to ensure the asymptotic stability of the IBF's. While we use different transformations for pairwise comparisons of the models, we also show that a single common transformation can be used when simultaneously comparing all three models. We then demonstrate that, when FBF's are used to compare these models, we may have to employ different, model-specific training fractions in order to achieve asymptotic stability of the FBF's.

APPLYING THE SAVAGE-DICKEY DENSITY RATIO TO DEFAULT BAYES FACTORS, WITH AN ILLUSTRATION TO OUTLIER DETECTION IN RANDOM EFFECTS MODELS

When no information is available and hence improper noninformative priors should be used, Bayes factor includes the unspecified constants and can not be calibrated. To solve this problem, we modify the intrinsic Bayes factor (IBF) of Berger and Pericchi 1-2 and the fractional Bayes factor (FBF) of O'Hagan [3] with the generalized Savage-Dickey density ratio of Verdinelli and Wasserman [4]. These modified IBF and FBF are applied to detecting outliers in random effects models with a mean-shift structure. The proposed methodology is exemplified by a simulation experiment with a generated data set and also applied to a real data set, Dyestuff data in Box and Tiao [5]

Consistent fractional Bayes factor for nested normal linear models

In the Bayesian approach to parametric model comparison, the use of improper priors is problematic due to the indeterminacy of the resulting Bayes factor (BF). The need for developing automatic and robust methods for model comparison has led to the introduction of alternative BFs. Intrinsic Bayes factors (Berger and Pericchi, 1996a) and fractional Bayes factors (FBF) (O'Hagan, 1995) are two alternative strategies for default model selection. We show in this paper that the FBF can be inconsistent. To overcome this problem, we propose a generalization of the FBF approach that leads to the usual FBF or to some variants of it in some special cases. As an important problem, we consider and discuss this generalization for model selection in nested linear models.

Bayesian assessment of goodness of fit against nonparametric alternatives

AbstractThe classical chi‐square test of goodness of fit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chi‐square test requires continuous data to be grouped into arbitrary categories. Furthermore, as the test is based upon an approximation, it can only be used if there are sufficient data. In practice, these requirements are often wasteful of information and overly restrictive. The authors explore the use of the fractional Bayes factor to obtain a Bayesian alternative to the chi‐square test when no specific prior information is available. They consider the extent to which their methodology can handle small data sets and continuous data without arbitrary grouping.

Sensitivity of the fractional Bayes factor to prior distributions

AbstractThe authors derive a measure of the sensitivity of the fractional Bayes factor, an index which is used to compare models when the priors for their respective parameters are improper, or when there is concern about robustness of the prior specification. They prove that in a large class of problems, this measure is a decreasing function of the fraction of the sample used to update the prior distribution before the models are compared.

Default Bayes factors for generalized linear models

Abstract Commonly used noninformative priors for generalized linear models (GLMs), such as the uniform prior or Jeffreys's prior, are generally improper, and thus are defined only up to arbitrary normalizing constants. Therefore, Bayes factors and posterior model probabilities are not well defined under these types of noninformative priors, making Bayesian hypothesis testing and model selection impossible. To overcome these difficulties, we derive the intrinsic Bayes factor (IBF) of Berger and Pericchi (1996a, In: Bayesian Analysis V. Bernardo, J.M. et al. (Eds.), Oxford University Press, Oxford, pp. 25–44; J. Amer. Statist. Assoc. 91, 109–122) and the fractional Bayes factor (FBF) of O'Hagan (1995, J. Roy Statist. Soc. Ser. B 57, 99–138) for the class of GLMs. Using the fact that the posterior distribution of the regression coefficients in a GLM is approximately normally distributed, we derive an asymptotic form for these default Bayes factors. We derive an explicit form of the IBF for the binary, Poisson, and exponential regression models. In addition, we derive criteria for obtaining the minimal training sample for this class of models. We demonstrate our results with two real datasets and a simulated dataset.

Bayes factors for a test about the drift of the Brownian motion under noninformative priors

Brownian motions are useful in modeling many stochastic phenomena. We address the problem of default testing for the sign of the drift, if any, in the mean of the process using the Bayesian approach. Conventional Bayes factors for hypotheses testing, however, cannot typically accommodate the use of standard noninformative priors, as such priors are defined only up to arbitrary constants which affect the values of the Bayes factors. To address this problem for some common noninformative priors, we shall use Intrinsic Bayes factors due to Berger and Pericchi (1996, J. Amer. Statist. Assoc. 91, 109–122) and fractional Bayes factors due to O'Hagan (1995, J. Roy. Statist. Soc. Ser. B 57(1), 99–138), assuming discrete observations are available from the process on a coarse time scale.

Methods for Default and Robust Bayesian Model Comparison: the Fractional Bayes Factor Approach

SummaryIn the Bayesian approach to model selection and hypothesis testing, the Bayes factor plays a central role. However, the Bayes factor is very sensitive to prior distributions of parameters. This is a problem especially in the presence of weak prior information on the parameters of the models. The most radical consequence of this fact is that the Bayes factor is undetermined when improper priors are used. Nonetheless, extending the non‐informative approach of Bayesian analysis to model selection/testing procedures is important both from a theoretical and an applied viewpoint. The need to develop automatic and robust methods for model comparison has led to the introduction of several alternative Bayes factors. In this paper we review one of these methods: the fractional Bayes factor (O'Hagan, 1995). We discuss general properties of the method, such as consistency and coherence. Furthermore, in addition to the original, essentially asymptotic justifications of the fractional Bayes factor, we provide further finite‐sample motivations for its use. Connections and comparisons to other automatic methods are discussed and several issues of robustness with respect to priors and data are considered. Finally, we focus on some open problems in the fractional Bayes factor approach, and outline some possible answers and directions for future research.

Methods for Default and Robust Bayesian Model Comparison: The Fractional Bayes Factor Approach

Methods for Default and Robust Bayesian Model Comparison: The Fractional Bayes Factor Approach

Default Bayes Factors for Nonnested Hypothesis Testing

Bayesian hypothesis testing for nonnested hypotheses is studied, using various “default” Bayes factors, such as the fractional Bayes factor, the median intrinsic Bayes factor, and the encompassing and expected intrinsic Bayes factors. The different default methods are first compared with each other and with the p value in normal one-sided testing, to illustrate the basic issues. General results for one-sided testing in location and scale models are then presented. The default Bayes factors are also studied for specific models involving multiple hypotheses. In particular, a multiple hypothesis testing example involving a sequential clinical trial is discussed. In most of the examples presented we also derive the intrinsic prior; this is the prior distribution, which, if used directly, would yield answers (asymptotically) equivalent to those for the given default Bayes factor.

Default Bayes Factors for Nonnested Hypothesis Testing

Abstract Bayesian hypothesis testing for nonnested hypotheses is studied, using various “default” Bayes factors, such as the fractional Bayes factor, the median intrinsic Bayes factor, and the encompassing and expected intrinsic Bayes factors. The different default methods are first compared with each other and with the p value in normal one-sided testing, to illustrate the basic issues. General results for one-sided testing in location and scale models are then presented. The default Bayes factors are also studied for specific models involving multiple hypotheses. In particular, a multiple hypothesis testing example involving a sequential clinical trial is discussed. In most of the examples presented we also derive the intrinsic prior; this is the prior distribution, which, if used directly, would yield answers (asymptotically) equivalent to those for the given default Bayes factor.

An Intrinsic Limiting Procedure for Model Selection and Hypotheses Testing

Improper priors typically arise in default Bayesian estimation problems. In the Bayesian approach to model selection or hypothesis testing, the main tool is the Bayes factor. When improper priors for the parameters appearing in the models are used, the Bayes factor is not well defined. The intrinsic Bayes factor introduced by Berger and Pericchi is an interesting method for overcoming that difficulty. That method is of particular interest as a means for generating proper prior distributions (intrinsic priors) for model comparison from the improper priors typically used in estimation. The goal of this article is to develop a limiting procedure that provides a solid justification for the use of Bayes factor with intrinsic priors. The procedure is formalized and discussed for nested and nonnested models. Illustrations and comparisons with other approximations to Bayes factors, such as the Bayesian information criterion of Schwarz and the fractional Bayes factor of O'Hagan are provided.

An Intrinsic Limiting Procedure for Model Selection and Hypotheses Testing

Abstract Improper priors typically arise in default Bayesian estimation problems. In the Bayesian approach to model selection or hypothesis testing, the main tool is the Bayes factor. When improper priors for the parameters appearing in the models are used, the Bayes factor is not well defined. The intrinsic Bayes factor introduced by Berger and Pericchi is an interesting method for overcoming that difficulty. That method is of particular interest as a means for generating proper prior distributions (intrinsic priors) for model comparison from the improper priors typically used in estimation. The goal of this article is to develop a limiting procedure that provides a solid justification for the use of Bayes factor with intrinsic priors. The procedure is formalized and discussed for nested and nonnested models. Illustrations and comparisons with other approximations to Bayes factors, such as the Bayesian information criterion of Schwarz and the fractional Bayes factor of O'Hagan are provided.

Bayesian analysis of autoregressive time series with change points

The paper deals with the identification of a stationary autoregressive model for a time series and the contemporary detection of a change in its mean. We adopt the Bayesian approach with weak prior information on the parameters of the models under comparison and an exact form of the likelihood function. When necessary, we resort to fractional Bayes factors to choose between models, and to importance sampling to solve computational issues.

Alternative Bayes factors for model selection

AbstractSeveral 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.

Properties of intrinsic and fractional Bayes factors

The Fractional Bayes Factor and various forms of Intrinsic Bayes Factor are related methods which have been proposed for Bayesian model comparison when prior information about model parameters is weak. This paper identifies and contrasts various properties of these methods, with particular reference to coherence and practicality.

Measuring Discordancy between Prior and Data

SUMMARY In this paper we consider possible measures of the degree of discordancy between a proper prior and observed data. A consideration of the problem when we gain complete knowledge of the parameters suggests that measures based on tail areas are unsuitable. A wish for invariance under transformations leads to a Bayes factor comparing the marginal distribution of the data with the proper prior with that with a non-informative prior. to overcome problems with improper priors some suggestions are compared. A measure based on a fractional Bayes factor is preferred. The method is illustrated for normal samples, binomial samples and linear models.

Fractional Bayes Factors for Model Comparison

SUMMARY Bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves. However, there are difficulties with this approach when prior information about the parameters of the various models is weak. Partial Bayes factors offer a resolution of the problem by setting aside part of the data as a training sample. The training sample is used to obtain an initial informative posterior distribution of the parameters in each model. Model comparison is then based on a Bayes factor calculated from the remaining data. Properties of partial Bayes factors are discussed, particularly in the context of weak prior information, and they are found to have advantages over other proposed methods of model comparison. A new variant of the partial Bayes factor, the fractional Bayes factor, is advocated on grounds of consistency, simplicity, robustness and coherence.