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

In 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.     

Model selection in toxicity studies

In toxicity studies, model mis‐specification could lead to serious bias or faulty conclusions. As a prelude to subsequent statistical inference, model selection plays a key role in toxicological studies. It is well known that the Bayes factor and the cross‐validation method are useful tools for model selection. However, exact computation of the Bayes factor is usually difficult and sometimes impossible and this may hinder its application. In this paper, we recommend to utilize the simple Schwarz criterion to approximate the Bayes factor for the sake of computational simplicity. To illustrate the importance of model selection in toxicity studies, we consider two real data sets. The first data set comes from a study of dietary fortification with carbonyl iron in which the Bayes factor and the cross‐validation are used to determine the number of sub‐populations in a mixture normal model. The second example involves a developmental toxicity study in which the selection of dose–response functions in a beta‐binomial model is explored.     

Prior‐free Bayes Factors Based on Data Splitting

Bayes factors that do not require prior distributions are proposed for testing one parametric model versus another. These Bayes factors are relatively simple to compute, relying only on maximum likelihood estimates, and are Bayes consistent at an exponential rate for nested models even when the smaller model is true. These desirable properties derive from the use of data splitting. Large sample properties, including consistency, of the Bayes factors are derived, and a simulation study explores practical concerns. The methodology is illustrated with civil engineering data involving compressive strength of concrete.     

An Analysis of the Value Added by Secondary Schools in England: Is the Value Added Indicator of Any Value?*

This paper argues that the value added score published for all publicly funded secondary schools in England is an unreliable indicator of school performance. A substantial proportion of the between‐school variation in the value added score is accounted for by factors outside the school's control. These factors include several pupil‐related variables such as the proportion of pupils on free school meals, the authorized absence rate of pupils and the proportion of pupils from ethnic minority backgrounds. The value added score is also related to several school characteristics such as the school's admission policy and its subject specialism. The main policy recommendation of this paper is that the value added score should not be used as a performance indicator, but should be used to gain a better understanding of why the value added score varies between schools.     

Bayesian model selection using encompassing priors

This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes.     

Bayesian Calibration of p‐Values from Fisher's Exact Test

p‐Values are commonly transformed to lower bounds on Bayes factors, so‐called minimum Bayes factors. For the linear model, a sample‐size adjusted minimum Bayes factor over the class of g‐priors on the regression coefficients has recently been proposed (Held & Ott, The American Statistician 70(4), 335–341, 2016). Here, we extend this methodology to a logistic regression to obtain a sample‐size adjusted minimum Bayes factor for 2 × 2 contingency tables. We then study the relationship between this minimum Bayes factor and two‐sided p‐values from Fisher's exact test, as well as less conservative alternatives, with a novel parametric regression approach. It turns out that for all p‐values considered, the maximal evidence against the point null hypothesis is inversely related to the sample size. The same qualitative relationship is observed for minimum Bayes factors over the more general class of symmetric prior distributions. For the p‐values from Fisher's exact test, the minimum Bayes factors do on average not tend to the large‐sample bound as the sample size becomes large, but for the less conservative alternatives, the large‐sample behaviour is as expected.     

Bayesian Model Selection with an Uninformative Prior*

Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill‐defined. Using careful consideration of the parameter of interest in cointegration analysis and a re‐specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283–306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with ‘ignorance’ priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model.     

On the natural selection rule in general linear models

We consider a problem of selecting the best treatment in a general linear model. We look at the properties of the natural selection rule. It is shown that the natural selection rule is minimax under to “0–1” loss function and it is a Bayes rule under a monotone permutation invariant loss function with respect to a permutation invariant prior for every variance balanced design. Some other condition on the design matrix is given so that a Bayes rule with respect to a normal prior will be of simple structure.     

Objective methods for graphical structural learning

Graphical models are used for expressing conditional independence relationships among variables by the means of graphs, whose structure is typically unknown and must be inferred by the data at hand. We propose a theoretically sound Objective Bayes procedure for graphical model selection. Our method is based on the Expected-Posterior Prior and on the Power-Expected-Posterior Prior. We use as input of the proposed methodology a default improper prior and suggest computationally efficient approximations of Bayes factors and posterior odds. In a variety of simulated scenarios with varying number of nodes and sample sizes, we show that our method is highly competitive with, or better than, current benchmarks. We also discuss an application to protein-signaling data, which wieldy confirms existing results in the scientific literature.     

Empirical Bayes analysis of log-linear models for a generalized finite stationary Markov chain

This article presents the empirical Bayes method for estimation of the transition probabilities of a generalized finite stationary Markov chain whose ith state is a multi-way contingency table. We use a log-linear model to describe the relationship between factors in each state. The prior knowledge about the main effects and interactions will be described by a conjugate prior. Following the Bayesian paradigm, the Bayes and empirical Bayes estimators relative to various loss functions are obtained. These procedures are illustrated by a real example. Finally, asymptotic normality of the empirical Bayes estimators are established.     

基于贝叶斯方法的项目投资风险决策

由于投资受政策、社会环境、经济环境、管理水平等诸多因素的制约，因而投资成败的不确定性极大。本文将贝叶斯决策理论应用于投资决策中，建立了投资贝叶斯风险决策模型，分析了决策模型中各种参数的确定方法，并阐述了该模型对降低决策风险的作用。在风险决策中，信息的价值可以定量，运用贝叶斯公式分析在风险决策中增大信息量，有益于降低决策风险。     

Recession forecasting using Bayesian classification

We demonstrate the use of a Naïve Bayes model as a recession forecasting tool. The approach is closely connected with Markov-switching models and logistic regression, but also has important differences. In contrast to Markov-switching models, our Naïve Bayes model treats National Bureau of Economic Research business cycle turning points as data, rather than as hidden states to be inferred by the model. Although Naïve Bayes and logistic regression are asymptotically equivalent under certain distributional assumptions, the assumptions do not hold for business cycle data. As a result, Naïve Bayes has a larger asymptotic error rate, but converges to the error rate more quickly than logistic regression, resulting in more accurate recession forecasts with limited data. We show that Naïve Bayes outperforms competing models and the Survey of Professional Forecasters consistently for real-time recession forecasting up to 12 months in advance. These results hold under standard error measures, and also under a novel measure that varies the penalty on false signals, depending on when they occur within a cycle; for example, a false signal in the middle of an expansion is penalized more heavily than one that occurs close to a turning point.     

Hybrid decision model for information project selection

Information project selection is an important and recurring activity in many organizations. Establishing a systematic project selection model thus is important for organizations. However, information system project selection is made difficult by the need to consider numerous quantitative and qualitative factors, such as business goals, benefits, project risks and limitations in available resources. This study proposes a hybrid measurement model for evaluating the overall effects of information projects. The selection model generates critical, quantitative and qualitative indicators to help managers realize the advantages/disadvantages of operational conditions. This hybrid selection model not only accelerates project evaluation and selection, but also helps organizations optimize project selection and thus achieve sustainable development.     

On the Risk Performance of Bays Empirical Bayes Procedures for Classification Between N(-1,l) and N(1,1)

In empirical Bayes decision making, the Bayes empirical Bayes approach is diccussed by Gilliland and Boyer (1979). In the finite state component case, the Bayes empirical Bayes procedures are shown to have optimal properties in a fairly general setting and believed to have small sample advantage over the classical rules. The flexibility of making desirable adjustments for these decision procedures by choice of prior enables one to set a proper strategy when dealing with actual problems.
The applications of Bayes empirical Bayes procedures, however, create some interesting theoretical and computational problems as they are fairly complicated in structure. This paper gives a brief introduction into the Bayes empirical Bayes approach, and, to illustrate it, explicit results are given for testing H₀: N(-1,1) against H₁: N(1,1).     

Performance of empirical Bayes estimators of random coefficients in multilevel analysis: Some results for the random intercept-only model

For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level-2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level-2 units. For large numbers of level-1 (30) or level-2 units (100), the performances of REML-based and ML-based EBEs are comparable. For small numbers of level-1 (10) and level-2 units (25), the REML-based EBEs have a lower Bayes risk than ML-based EBEs only for high intraclass correlations (0.5).     

Feature weighted naïve Bayes algorithm for information retrieval of enterprise systems

Automated information retrieval is critical for enterprise information systems to acquire knowledge from the vast amount of data sets. One challenge in information retrieval is text classification. Current practices rely heavily on the classical naïve Bayes algorithm due to its simplicity and robustness. However, results from this algorithm are not always satisfactory. In this article, the limitations of the naïve Bayes algorithm are discussed, and it is found that the assumption on the independence of terms is the main reason for an unsatisfactory classification in many real-world applications. To overcome the limitations, the dependent factors are considered by integrating a term frequency–inverse document frequency (TF-IDF) weighting algorithm in the naïve Bayes classification. Moreover, the TF-IDF algorithm itself is improved so that both frequencies and distribution information are taken into consideration. To illustrate the effectiveness of the proposed method, two simulation experiments were conducted, and the comparisons with other classification methods have shown that the proposed method has outperformed other existing algorithms in terms of precision and index recall rate.     

Data‐Driven Identification Constraints for DSGE Models

We propose imposing data‐driven identification constraints to alleviate the multimodality problem arising in the estimation of poorly identified dynamic stochastic general equilibrium models under non‐informative prior distributions. We also devise an iterative procedure based on the posterior density of the parameters for finding these constraints. An empirical application to the Smets and Wouters ( 2007 ) model demonstrates the properties of the estimation method, and shows how the problem of multimodal posterior distributions caused by parameter redundancy is eliminated by identification constraints. Out‐of‐sample forecast comparisons as well as Bayes factors lend support to the constrained model.     

Measuring location selection factors for international resort parks

This study presents a framework of issues to analyze Porter’s (The Competitive Advantage of Nations, 1990) Diamond model, and develops factors for determining the optimality of an international resort park location using the modified Delphi model. A panel of 16 experts from various backgrounds, including academia, government and business, provided input for the selection of location factors. Following three discussions, panel members reached consensus and selected the following set of 26 factors for optimizing location selection for international resort parks. This study also provides direction for applying the proposed model and suggestions for future research.     

Filipino ethnic entrepreneurship: An integrated review and propositions

With a large percentage of its population as overseas workers or emigrants, a number now in the millions and likely grow even larger, an examination is appropriate of the entrepreneurial challenges and opportunities faced by emigrants from the Philippines. This study explores the factors that shape entrepreneurship among emigrants in general and, in particular, among Filipinos in the United States of America. Integrating literature from a variety of sources relating to the migration of Filipinos and the creation and management of new ventures by ethnic entrepreneurs in the United States, it proposes a general model of ethnic entrepreneurship, develops propositions relating to the propensity of Filipino emigrants to establish entrepreneurial ventures, and concludes with suggestions for future research.     

An improved empirical Bayes test for positive exponential families

We exhibit an empirical Bayes test δ^* _n for a decision problem using a linear error loss in a class of positive exponential families. This empirical Bayes test δ^* _n possesses the asymptotic optimality, and its associated regret converges to zero with rate n ⁻¹(ln n )⁶ This rate of convergence improves the previous results in the literature in the sense that a faster rate of convergence is achieved under much weaker conditions. Examples are presented to illustrate the performance of the empirical Bayes test δ^* _n