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.     

A Bayesian estimation of lag lengths in distributed lag models

Dynamic regression models are widely used because they express and model the behaviour of a system over time. In this article, two dynamic regression models, the distributed lag (DL) model and the autoregressive distributed lag model, are evaluated focusing on their lag lengths. From a classical statistics point of view, there are various methods to determine the number of lags, but none of them are the best in all situations. This is a serious issue since wrong choices will provide bad estimates for the effects of the regressors on the response variable. We present an alternative for the aforementioned problems by considering a Bayesian approach. The posterior distributions of the numbers of lags are derived under an improper prior for the model parameters. The fractional Bayes factor technique [A. O'Hagan, Fractional Bayes factors for model comparison (with discussion), J. R. Statist. Soc. B 57 (1995), pp. 99–138] is used to handle the indeterminacy in the likelihood function caused by the improper prior. The zero-one loss function is used to penalize wrong decisions. A naive method using the specified maximum number of DLs is also presented. The proposed and the naive methods are verified using simulation data. The results are promising for the method we proposed. An illustrative example with a real data set is provided.     

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.     

Objective Bayesian multiple comparisons for normal variances

This paper considers the multiple comparisons problem for normal variances. We propose a solution based on a Bayesian model selection procedure to this problem in which no subjective input is considered. We construct the intrinsic and fractional priors for which the Bayes factors and model selection probabilities are well defined. The posterior probability of each model is used as a model selection tool. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and some frequentist tests.     

Intrinsic Priors for Testing Two Normal Means with Intrinsic Bayes Factors

ABSTRACT

In this article we consider the problem of comparing two normal means with unknown common variance using a Bayesian approach. Conventional Bayes factors with improper non informative priors are not well defined. The intrinsic Bayes factors are used to overcome such a difficulty. We derive intrinsic priors whose Bayes factors are asymptotically equivalent to the corresponding intrinsic Bayes factors. We illustrate our results with numerical examples.     

Objective priors for hypothesis testing in one-way random effects models

The Bayes factor is a key tool in hypothesis testing. Nevertheless, the important issue of which priors should be used to develop objective Bayes factors remains open. The authors consider this problem in the context of the one-way random effects model. They use concepts such as orthogonality, predictive matching and invariance to justify a specific form of the priors for common parameters and derive the intrinsic and divergence based prior for the new parameter. The authors show that both intrinsic priors or divergence-based priors produce consistent Bayes factors. They illustrate the methods and compare them with other proposals.     

Intrinsic Priors for Model Selection Using an Encompassing Model with Applications to Censored Failure Time Data

In Bayesian model selection or testingproblems one cannot utilize standard or default noninformativepriors, since these priors are typically improper and are definedonly up to arbitrary constants. Therefore, Bayes factors andposterior probabilities are not well defined under these noninformativepriors, making Bayesian model selection and testing problemsimpossible. We derive the intrinsic Bayes factor (IBF) of Bergerand Pericchi (1996a, 1996b) for the commonly used models in reliabilityand survival analysis using an encompassing model. We also deriveproper intrinsic priors for these models, whose Bayes factors are asymptoticallyequivalent to the respective IBFs. We demonstrate our resultsin three examples.     

Objective Bayesian Testing of a Poisson Mean

We consider testing hypotheses about a single Poisson mean. When prior information is not available, use of objective priors is of interest. We provide intrinsic priors based on the arithmetic intrinsic and fractional Bayes factors, and evaluate their characteristics.     

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.     

Posterior robustness with more than one sampling model

In order to robustify posterior inference, besides the use of large classes of priors, it is necessary to consider uncertainty about the sampling model. In this article we suggest that a convenient and simple way to incorporate model robustness is to consider a discrete set of competing sampling models, and combine it with a suitable large class of priors. This set reflects foreseeable departures of the base model, like thinner or heavier tails or asymmetry. We combine the models with different classes of priors that have been proposed in the vast literature on Bayesian robustness with respect to the prior. Also we explore links with the related literature of stable estimation and precise measurement theory, now with more than one model entertained. To these ends it will be necessary to introduce a procedure for model comparison that does not depend on an arbitrary constant or scale. We utilize a recent development on automatic Bayes factors with self-adjusted scale, the ‘intrinsic Bayes factor’ (Berger and Pericchi, Technical Report, 1993).     

Default Bayesian analysis of the Behrens–Fisher problem

In the Bayesian approach, the Behrens–Fisher problem has been posed as one of estimation for the difference of two means. No Bayesian solution to the Behrens–Fisher testing problem has yet been given due, perhaps, to the fact that the conventional priors used are improper. While default Bayesian analysis can be carried out for estimation purposes, it poses difficulties for testing problems. This paper generates sensible intrinsic and fractional prior distributions for the Behrens–Fisher testing problem from the improper priors commonly used for estimation. It allows us to compute the Bayes factor to compare the null and the alternative hypotheses. This default procedure of model selection is compared with a frequentist test and the Bayesian information criterion. We find discrepancy in the sense that frequentist and Bayesian information criterion reject the null hypothesis for data, that the Bayes factor for intrinsic or fractional priors do not.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

Bayesian approach to change point problems

A Bayesian approach is considered to study the change point problems. A hypothesis for testing change versus no change is considered using the notion of predictive distributions. Bayes factors are developed for change versus no change in the exponential families of distributions with conjugate priors. Under vague prior information, both Bayes factors and pseudo Bayes factors are considered. A new result is developed which describes how the overall Bayes factor has a decomposition into Bayes factors at each point. Finally, an example is provided in which the computations are performed using the concept of imaginary observations.     

Compatible prior distributions for directed acyclic graph models

Summary.  The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given directed acyclic graph model, using a conditioning approach. We define a class of parameterizations that is consistent with the modular structure of the directed acyclic graph and derive a procedure, that is invariant within this class, which we name reference conditioning.     

Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

A Comparison of Bayes Factors for Separated Models: Some Simulation Results

Some alternative Bayes Factors: Intrinsic, Posterior, and Fractional have been proposed to overcome the difficulties presented when prior information is weak and improper prior are used. Additional difficulties also appear when the models are separated or non nested. This article presents both simulation results and some illustrative examples analysis comparing these alternative Bayes factors to discriminate among the Lognormal, the Weibull, the Gamma, and the Exponential distributions. Simulation results are obtained for different sample sizes generated from the distributions. Results from simulations indicates that these alternative Bayes factors are useful for comparing non nested models. The simulations also show some similar behavior and that when both models are true they choose the simplest model. Some illustrative example are also presented.     

Some Faults of the Bayes Factor in Nonparametric Model Selection

We compare different Bayesian strategies for testing a parametric model versus a nonparametric alternative on the ground of their ability to solve the inconsistency problems arising when using the Bayes factor under certain conditions. A preliminary critical discussion of such an inconsistency is provided.     

A bayesian approach for testing periodicity of dichotomous observations

A Bayesian approach is utilized to test for periodicity in a dichotomous time series. Dichotomous data arise in a variety of circumstances when a variable takes on only two possible values. Conjugate and noninformative priors are considered as well as a hierarchical Bayes approach; the latter is considered the superior Bayes methodology. The situation of stochastic period lengths is also discussed. The generalization to the multinomial model is investigated to allow for the case that a variable takes on more than two possible values. In all cases decisions are made based on a Bayes factor. The proposed procedures are demonstrated on earthquake data in the central Virginia seismic zone     

A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.