Subjectivism has become the dominant philosophical foundation for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called “noninformative” priors, that is, priors constructed by some formal rule. We review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used. We give special emphasis to Jeffreys's rules and discuss the evolution of his viewpoint about the interpretation of priors, away from unique representation of ignorance toward the notion that they should be chosen by convention. We conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly: When sample sizes are small (relative to the number of parameters being estimated), it is dangerous to put faith in any “default” solution; but when asymptotics take over, Jeffreys's rules and their variants remain reasonable choices. We also provide an annotated b...

The Selection of Prior Distributions by Formal Rules

Elicitation is a key task for subjectivist Bayesians. Although skeptics hold that elicitation cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subject-matter expert colleagues. This article reviews the state of the art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful; that is, what criteria should be used. Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily “true” in some objectivistic sense, and cannot be judged in that way. We see that elicitation as simply part of the process of statistical modeling. Indeed, in a hierarchical model at which point the likelihood ends and the pr...

Statistical Methods for Eliciting Probability Distributions

: Scientific learning is an iterative process employing Criticism and Estimation. Correspondingly the formulated model factors into two complimentary parts - a predictive part allowing model criticism, and a Bayes posterior part allowing estimation. Implications for significance tests, the theory of precise measurement, and for ridge estimates are considered. Predictive checking functions for transformation, serial correlation, bad values, and their relation with Bayesian options are considered. Robustness is seen from a Bayesian viewpoint and examples are given. For the bad value problem a comparison with M estimators is made. (Author)

Sampling and Bayes' inference in scientific modelling and robustness

P values are the most commonly used tool to measure evidence against a hypothesis or hypothesized model. Unfortunately, they are often incorrectly viewed as an error probability for rejection of the hypothesis or, even worse, as the posterior probability that the hypothesis is true. The fact that these interpretations can be completely misleading when testing precise hypotheses is first reviewed, through consideration of two revealing simulations. Then two calibrations of a ρ value are developed, the first being interpretable as odds and the second as either a (conditional) frequentist error probability or as the posterior probability of the hypothesis.

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Calibration of p Values for Testing Precise Null Hypotheses

Survey Kernel methods in system identification, machine learning and function estimation: A survey

Robust 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.

An overview of robust Bayesian analysis

Robust Bayesian Analysis with ϵ‐Contaminations Partially Known

The visual behaviour is a determining factor in sailing due to the influence of the environmental conditions. The aim of this research was to determine the visual behaviour pattern in sailors with different practice time in one star race, applying a probabilistic model based on Markov chains. The sample of this study consisted of 20 sailors, distributed in two groups, top ranking (n = 10) and bottom ranking (n = 10), all of them competed in the Optimist Class. An automated system of measurement, which integrates the VSail-Trainer sail simulator and the Eye Tracking System(TM) was used. The variables under consideration were the sequence of fixations and the fixation recurrence time performed on each location by the sailors. The event consisted of one of simulated regatta start, with stable conditions of wind, competitor and sea. Results show that top ranking sailors perform a low recurrence time on relevant locations and higher on irrelevant locations while bottom ranking sailors make a low recurrence time in most of the locations. The visual pattern performed by bottom ranking sailors is focused around two visual pivots, which does not happen in the top ranking sailor's pattern. In conclusion, the Markov chains analysis has allowed knowing the visual behaviour pattern of the top and bottom ranking sailors and its comparison.

A probabilistic model for analysing the effect of performance levels on visual behaviour patterns of young sailors in simulated navigation

Approximation of the posterior density for diffusion processes

In many statistical problems we deal with more than one model. When the prior information on the parameters of the models is vague default priors are typically used. Unfortunately, these priors are usually improper provoking a calibration problem which precludes the comparison of the models. An attempt for solving this difficulty consists in using intrinsic priors, introduced in Berger and Pericchi (1996, The intrinsic Bayes factor for model selection and prediction. J Am Stat Assoc 91:109–122), instead of the original default priors; however, there are situations where the class of intrinsic priors is too large.

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Juan Antonio Cano

Papers

An overview of robust Bayesian analysis

Robust Bayesian Analysis with ϵ‐Contaminations Partially Known

A probabilistic model for analysing the effect of performance levels on visual behaviour patterns of young sailors in simulated navigation

Approximation of the posterior density for diffusion processes

Integral equation solutions as prior distributions for Bayesian model selection