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James O. Berger

Publications and source records attributed to James O. Berger.

At least 19 recordsLinked to original sources
arXiv2026

RobustGaSP: Robust Gaussian Stochastic Process Emulation in R

Gaussian stochastic process emulation is a powerful tool for approximating computationally intensive computer models. However, estimation of parameters in the GaSP emulator is a challenging task. No closed-form estimator is available, and many numerical problems arise with standard estimates, e.g., the maximum likelihood estimator. In this package, we implement a marginal posterior mode estimator for special priors and parameterizations, an estimation method that meets robust parameter estimation criteria; mathematical reasons are provided therein to explain why robust parameter estimation can greatly improve predictive performance of the emulator. In addition, inert inputs (inputs that almost have no effect on the variability of a function) can be identified from the marginal posterior mode estimation, at no extra computational cost. The package also implements the parallel partial Gaussian stochastic process (PP GaSP) emulator for scenarios where computer models have multiple outputs on e.g., spatio-temporal coordinates. The package can be operated in a default mode, but also allows numerous user specifications, such as the capability of specifying trend functions and noise terms. Examples are studied herein to highlight the performance of the package in terms of out-of-sample prediction.

stat.CO↗
arXiv2019

Restricted type II maximum likelihood priors on regression coefficients

In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (Lindley's paradox). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are "estimated" by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding restrictions which ensure that the resulting priors are at least as vague as conventional proper priors for model uncertainty. We find that these type II ML priors typically yield results that are close to answers obtained with the Bayesian Information Criterion (BIC).

math.ST↗
arXiv2018

The Median Probability Model and Correlated Variables

The median probability model (MPM) Barbieri and Berger (2004) is defined as the model consisting of those variables whose marginal posterior probability of inclusion is at least 0.5. The MPM rule yields the best single model for prediction in orthogonal and nested correlated designs. This result was originally conceived under a specific class of priors, such as the point mass mixtures of non-informative and g-type priors. The MPM rule, however, has become so very popular that it is now being deployed for a wider variety of priors and under correlated designs, where the properties of MPM are not yet completely understood. The main thrust of this work is to shed light on properties of MPM in these contexts by (a) characterizing situations when MPM is still safe under correlated designs, (b) providing significant generalizations of MPM to a broader class of priors (such as continuous spike-and-slab priors). We also provide new supporting evidence for the suitability of g-priors, as opposed to independent product priors, using new predictive matching arguments. Furthermore, we emphasize the importance of prior model probabilities and highlight the merits of non-uniform prior probability assignments using the notion of model aggregates.

math.ST↗
arXiv2018

Bayesian Methods for Analysis and Adaptive Scheduling of Exoplanet Observations

We describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian data analysis tools for extrasolar planet (exoplanet) detection, planetary orbit estimation, and adaptive scheduling of observations. Our work addresses analysis of stellar reflex motion data, where a planet is detected by observing the "wobble" of its host star as it responds to the gravitational tug of the orbiting planet. Newtonian mechanics specifies an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions; it is even more complex when multiple planets are present. The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky). Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis. This relies on the model being linearly separable, enabling partial analytical marginalization, reducing the dimension of the parameter space. Subsequent analysis uses adaptive Markov chain Monte Carlo methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations). We present an overview of our current techniques and highlight directions being explored by ongoing research.

astro-ph.IM↗
arXiv2017

A note on recent criticisms to Birnbaum's theorem

In this note, we provide critical commentary on two articles that cast doubt on the validity and implications of Birnbaum's theorem: Evans (2013) and Mayo (2014). In our view, the proof is correct and the consequences of the theorem are alive and well.

math.ST↗
arXiv2017

On the Ubiquity of Information Inconsistency for Conjugate Priors

Informally, "Information Inconsistency" is the property that has been observed in many Bayesian hypothesis testing and model selection procedures whereby the Bayesian conclusion does not become definitive when the data seems to become definitive. An example is that, when performing a t-test using standard conjugate priors, the Bayes factor of the alternative hypothesis to the null hypothesis remains bounded as the t statistic grows to infinity. This paper shows that information inconsistency is ubiquitous in Bayesian hypothesis testing under conjugate priors. Yet the title does not fully describe the paper, since we also show that theoretically recommended priors, including scale mixtures of conjugate priors and adaptive priors, are information consistent. Hence the paper is simply a forceful warning that use of conjugate priors in testing and model selection is highly problematical, and should be replaced by the information consistent alternatives.

math.ST↗
arXiv2017

Robust Gaussian Stochastic Process Emulation

We consider estimation of the parameters of a Gaussian Stochastic Process (GaSP), in the context of emulation (approximation) of computer models for which the outcomes are real-valued scalars. The main focus is on estimation of the GaSP parameters through various generalized maximum likelihood methods, mostly involving finding posterior modes; this is because full Bayesian analysis in computer model emulation is typically prohibitively expensive. The posterior modes that are studied arise from objective priors, such as the reference prior. These priors have been studied in the literature for the situation of an isotropic covariance function or under the assumption of separability in the design of inputs for model runs used in the GaSP construction. In this paper, we consider more general designs (e.g., a Latin Hypercube Design) with a class of commonly used anisotropic correlation functions, which can be written as a product of isotropic correlation functions, each having an unknown range parameter and a fixed roughness parameter. We discuss properties of the objective priors and marginal likelihoods for the parameters of the GaSP and establish the posterior propriety of the GaSP parameters, but our main focus is to demonstrate that certain parameterizations result in more robust estimation of the GaSP parameters than others, and that some parameterizations that are in common use should clearly be avoided. These results are applicable to many frequently used covariance functions, e.g., power exponential, Mat{é}rn, rational quadratic and spherical covariance. We also generalize the results to the GaSP model with a nugget parameter. Both theoretical and numerical evidence is presented concerning the performance of the studied procedures.

math.ST↗
arXiv2016

Comparison of Bayesian and Frequentist Multiplicity Correction For Testing Mutually Exclusive Hypotheses Under Data Dependence

The problem of testing mutually exclusive hypotheses with dependent test statistics is considered. Bayesian and frequentist approaches to multiplicity control are studied and compared to help gain understanding as to the effect of test statistic dependence on each approach. The Bayesian approach is shown to have excellent frequentist properties and is argued to be the most effective way of obtaining frequentist multiplicity control, without sacrificing power, when there is considerable test statistic dependence.

math.ST↗
arXiv2015

Rejection Odds and Rejection Ratios: A Proposal for Statistical Practice in Testing Hypotheses

Much of science is (rightly or wrongly) driven by hypothesis testing. Even in situations where the hypothesis testing paradigm is correct, the common practice of basing inferences solely on p-values has been under intense criticism for over 50 years. We propose, as an alternative, the use of the odds of a correct rejection of the null hypothesis to incorrect rejection. Both pre-experimental versions (involving the power and Type I error) and post-experimental versions (depending on the actual data) are considered. Implementations are provided that range from depending only on the p-value to consideration of full Bayesian analysis. A surprise is that all implementations -- even the full Bayesian analysis -- have complete frequentist justification. Versions of our proposal can be implemented that require only minor modifications to existing practices yet overcome some of their most severe shortcomings.

stat.ME↗
arXiv2015

Overall Objective Priors

In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a single objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior.

math.ST↗
arXiv2013

Bayesian analysis of dynamic item response models in educational testing

Item response theory (IRT) models have been widely used in educational measurement testing. When there are repeated observations available for individuals through time, a dynamic structure for the latent trait of ability needs to be incorporated into the model, to accommodate changes in ability. Other complications that often arise in such settings include a violation of the common assumption that test results are conditionally independent, given ability and item difficulty, and that test item difficulties may be partially specified, but subject to uncertainty. Focusing on time series dichotomous response data, a new class of state space models, called Dynamic Item Response (DIR) models, is proposed. The models can be applied either retrospectively to the full data or on-line, in cases where real-time prediction is needed. The models are studied through simulated examples and applied to a large collection of reading test data obtained from MetaMetrics, Inc.

stat.AP↗
arXiv2010

Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham's-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

math.ST↗
arXiv2009

The formal definition of reference priors

Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically.

math.ST↗
arXiv2008

Objective Bayes testing of Poisson versus inflated Poisson models

The Poisson distribution is often used as a standard model for count data. Quite often, however, such data sets are not well fit by a Poisson model because they have more zeros than are compatible with this model. For these situations, a zero-inflated Poisson (ZIP) distribution is often proposed. This article addresses testing a Poisson versus a ZIP model, using Bayesian methodology based on suitable objective priors. Specific choices of objective priors are justified and their properties investigated. The methodology is extended to include covariates in regression models. Several applications are given.

math.ST↗
arXiv2008

Objective Bayesian analysis under sequential experimentation

Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered.

math.ST↗
arXiv2008

Objective priors for the bivariate normal model

Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investigation, a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, including the correlation coefficient.

math.ST↗