Juan Antonio Cano

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.

On Intrinsic Priors for Nonnested Models

Model selection problems involving nonnested models are considered. Bayes factor based solution to these problems needs prior distributions for the parameters in the alternative models. When the prior information on these parameters is vague default priors are available but, unfortunately, these priors are usually imporper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. Intrinsic priors have been introduced for solving this difficulty. While these priors are well established for nested models, their construction for nonnested models is still an open problem.In this latter setting this paper studies the system of functional equations that defines the intrinsic priors. It is shown that the solutions to these equations are obtained from the solutions to a single homogeneous linear functional equation. The Bayes factors associated with these solutions are analyzed. Some illustrative examples are provided and, in particular, location, scale, and location-scale models are considered.

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

Abstract 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 SystemTM 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 sailors pattern. In conclusion, the Markov chains analysis has allowed knowing the visual behaviour pattern of the top and bottom ranking sailors and its comparison.

Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X1,X2, … ,Xp)τ be a random vector, the X1 being independently distributed as N(θ1,σ2) random variables (σ2 known), while the θ1 are thought to be exchangeable, modelled as i.i.d, N(μ,τ2). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ2 is assumed to be independent of µ having a distribution g(τ2) lying in a certain class of distributions g. For several gs, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ1 as g ranges over g.

Classes of bidimensional priors specified on a collection of set: Bayesian robustness

When the parameter space is multidimensional, to elicit the joint prior distribution is a very difficult task. An accessible prior information might then be the class of prior distributions with given one-dimensional marginals. Unfortunately, even in bidimensional parameter spaces, the variational problems encountered in the Bayesian analysis of this class have not yet been solved. This paper is devoted to studying two approximations to these variational problem based on the observation that the above class of priors can be seen as the class of priors with specified probabilities for a given (non-countable) collection of sets. Illustrations, including a recent clinical trial (ECMO), are given.

Objective Bayesian Analysis of an Exponential Regression Model with Constrained Parameters Applied to Animal Digestibility

In animal digestibility the proportion of degraded food along the time has usually been modeled as a normal random variable with mean a function of the time and the following three parameters: the proportion of degraded food almost instantaneously, remaining proportion of food to be degraded, and velocity of degradation. The estimation of these parameters has been carried out mainly from a frequentist viewpoint by using the asymptotic distribution of the maximum likelihood estimator. This may give inadmissible estimates, such as values outside of the range of the parameters. This drawback could not appear if a Bayesian approach were adopted. In this article an objective Bayesian analysis is developed and illustrated on real and simulated data.

Reducing Monte Carlo error in the Bayesian estimation of risk ratios using log-binomial regression models

In cohort studies, binary outcomes are very often analyzed by logistic regression. However, it is well known that when the goal is to estimate a risk ratio, the logistic regression is inappropriate if the outcome is common. In these cases, a log-binomial regression model is preferable. On the other hand, the estimation of the regression coefficients of the log-binomial model is difficult owing to the constraints that must be imposed on these coefficients. Bayesian methods allow a straightforward approach for log-binomial regression models and produce smaller mean squared errors in the estimation of risk ratios than the frequentist methods, and the posterior inferences can be obtained using the software WinBUGS. However, Markov chain Monte Carlo methods implemented in WinBUGS can lead to large Monte Carlo errors in the approximations to the posterior inferences because they produce correlated simulations, and the accuracy of the approximations are inversely related to this correlation. To reduce correlation and to improve accuracy, we propose a reparameterization based on a Poisson model and a sampling algorithm coded in R.

Objective Bayesian vs. least squares estimation for by-products degradability with different rumen fluids.

The degradation kinetic curves of different by-products have been obtained. The considered by-products were lemon and several types of treated and untreated barley straw, and they were degraded by in vitro incubation with rumen fluid extracted from two herds of Murciano-Granadina goats, one of them fed alfalfa hay and the other one fed barley straw. The feeds were incubated at 39oC for 12, 24, 36, 48 and 72 hours with each rumen fluid. The resulting fitted exponential-type degradation curves obtained with a frequentist statistical analysis were compared with those resulting from an objective Bayesian statistical analysis. The use of the objective Bayesian analysis smoothed the estimates of the frequentist fit using least squares, which did not suitably process the involved restrictions and avoided biologically unacceptable results. On the other hand, the rumen fluid from goats fed alfalfa hay fomented the greatest effective degradability and the degradabilities of the different by-products were also compa...