José M. Bernardo

An overview of robust Bayesian analysis

SummaryRobust Bayesian analysis is the study of the sensitivity of Bayesian answers to uncertain inputs. This paper seeks to provide an overview of the subject, one that is accessible to statisticians outside the field. Recent developments in the area are also reviewed, though with very uneven emphasis.

Estimating a Product of Means: Bayesian Analysis with Reference Priors

Abstract Suppose that we observe X ∼ N(α, 1) and, independently, Y ∼ N(β, 1), and are concerned with inference (mainly estimation and confidence statements) about the product of means θ = αβ. This problem arises, most obviously, in situations of determining area based on measurements of length and width. It also arises in other practical contexts, however. For instance, in gypsy moth studies, the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass. Approximately independent samples can be obtained for each mean (see Southwood 1978). Noninformative prior Bayesian approaches to the problem are considered, in particular the reference prior approach of Bernardo (1979). An appropriate reference prior for the problem is developed, and relatively easily implementable formulas for posterior moments (e.g., the posterior mean and variance) and credible sets are derived. Comparisons with alternative noninf...

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.

Scoring rules and the evaluation of probabilities

SummaryIn Bayesian inference and decision analysis, inferences and predictions are inherently probabilistic in nature. Scoring rules, which involve the computation of a score based on probability forecasts and what actually occurs, can be used to evaluate probabilities and to provide appropriate incentives for “good” probabilities. This paper review scoring rules and some related measures for evaluating probabilities, including decompositions of scoring rules and attributes of “goodness” of probabilites, comparability of scores, and the design of scoring rules for specific inferential and decision-making problems

Reference Priors in a Variance Components Problem

The ordered group reference prior algorithm of Berger and Bernardo (1989b) is applied to the balanced variance components problem. Besides the intrinsic interest of developing good noninformative priors for the variance components problem, a number of theoretically interesting issues arise in application of the proposed procedure. The algorithm is described (for completeness) in an important special case, with a detailed heuristic motivation.

An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters

This paper offers an introduction to Bayesian reference analysis, often described as the more successful method to produce non-subjective, model-based, posterior distributions. The ideas are illustrated in detail with an interesting problem, the ratio of multinomial parameters, for which no model-based Bayesian analysis has been proposed. Signposts are provided to the huge related literature.

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.

Bayesian Statistics 9

We thank all of the discussants for their valuable insights and elaborations. In particular, we thank Prof. Clarke and Dr. Severinski for their conjectured extension to Theorem 3, the product of many personal discussions both in Austin and in Spain (and probably many more hours of work in Miami). The conjecture seems quite likely to be true, and strikes us as a nice way of understanding adaptive penalty functions and infinite-dimensional versions of the corresponding shrinkage priors. Rather than respond to each of the six discussions in turn, we have grouped the comments into three rough categories.I describe ongoing work on development of Bayesian methods for exploring periodically varying phenomena in astronomy, addressing two classes of sources: pulsars, and extrasolar planets (exoplanets). For pulsars, the methods aim to detect and measure periodically varying signals in data consisting of photon arrival times, modeled as non-homogeneous Poisson point processes. For exoplanets, the methods address detection and estimation of planetary orbits using observations of the reflex motion “wobble” of a host star, including adaptive scheduling of observations to optimize inferences.

Exponential and bayesian conjugate families: Review and extensions

SummaryThe notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting features. Relationships between such classes are also discussed. Our aim is to provide a unified approach to the theory of conjugate families for exponential family likelihoods. An important aspect of the theory concerns reparameterisations of the exponential family under consideration. We briefly review the concept of a conjugate parameterisation, which provides further insight into many of the properties discussed throughout the paper. Finally, further implications of these results for Bayesian conjugate analysis of exponential families are investigated.

Objective Priors for Discrete Parameter Spaces

This article considers the development of objective prior distributions for discrete parameter spaces. Formal approaches to such development—such as the reference prior approach—often result in a constant prior for a discrete parameter, which is questionable for problems that exhibit certain types of structure. To take advantage of structure, this article proposes embedding the original problem in a continuous problem that preserves the structure, and then using standard reference prior theory to determine the appropriate objective prior. Four different possibilities for this embedding are explored, and applied to a population-size model, the hypergeometric distribution, the multivariate hypergeometric distribution, the binomial-beta distribution, and the binomial distribution. The recommended objective priors for the first, third, and fourth problems are new.