Julián de la Horra

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.

Posterior predictive p-values: what they are and what they are not

Schervish (1996) carried out an interesting study about some properties of the classicalp-value. In this paper, a similar study is carried out for posterior predictivep-values, in a general setting, showing that: a) the posterior predictivep-value is a continuous function of the null hypothesis, for fixed data, b) the posterior predictivep-value cannot be interpreted (in general) as a measure of support for the null hypothesis.

Asymptotic behaviour of the posterior predictive p-value

The posterior predictive p-value is a Bayesian-motivated alternative to the classical concept of p-value. This paper is devoted to study its asymptotic behaviour. Under mild assumptions, it is proved that: a)When θ*, the true value of the parameter, belongs to the null hypothesis, the distribution of the posterior predictive p-value converges to the uniform distribution over the interval (0,1). b)When θ* does not belong to the null hypothesis, the posterior predictive p-value converges almost surely to zero

The posterior predictive p-value for the problem of goodness of fit

The aim of this paper is to explore some features and possible uses of the posterior predictivep-value for the problem of goodness of fit. First, the behaviour of the posterior predictivep-value is compared with the behaviour of the classicalp-value through some interesting examples. Then, we consider a decision problem for simultaneously deciding to accept/reject a modelM and to accept/reject a null hypothesis (if we have accepted the modelM); the posterior predictivep-value is used for estimating the posterior probability of the model.

Bayesian analysis under ε-contaminated priors: a trade-off between robustness and precision

Abstract Wasserman (1989) proved that the γ-level highest likelihood region is robust in the sense that its posterior probability content is least sensitive, when the prior goes through an e-contamination class. First, Wassermans result is slightly improved upon (less restrictive assumptions, a simpler proof,…). Then, a credible region is considered as an estimate of the parameter; if the most robust region is searched in the class of sets with posterior probability content greater than or equal to γ0, we find that all the precision for estimating the parameter θ is lost; so, it seems sensible to consider the class of credible regions with a posterior probability level at least equal to γ0 and a certain precision (Lebesgue measure at most l0); if Θ is an open interval in R , a method for obtaining the most robust interval (in this class) is developed.

Bayesian Robustness of the Posterior Predictive p-Value

Abstract In this paper, the Bayesian robustness of the posterior predictive p-value is studied. First of all, it is proved that Lavines linearization technique can be extended for analyzing this problem. Then, the result is applied to the ϵ-contamination class of prior distributions.Abstract In this paper, the Bayesian robustness of the posterior predictive p-value is studied. First of all, it is proved that Lavines linearization technique can be extended for analyzing this problem. Then, the result is applied to the ϵ-contamination class of prior distributions.

Optimality of the posterior predictive p-value based on the posterior. Odds

Thompson (1997) considered a wide definition of p-value and found the Baves p-value for testing a ooint null hypothesis H0: θ= θ0 versus H1: θ ≠ θ0. In this paper, the general case of testing H0: θ ∈ ⊝0 versus H1: θ ∈ ⊝c 0 is studied. A generalization of the concept of p-value is given, and it is proved that the posterior predictive p-value based on the posterior odds is (asymptotically) a Bayes p-value. Finally, it is suggested that this posterior predictive p-value could be used as a reference p-value

Bayesian Model Selection: Measuring the χ2 Discrepancy with the Uniform Distribution

In the last years, many articles have been written about Bayesian model selection. In this article, a different and easier method is proposed and analyzed. The key idea of this method is based on the well-known property that, under the true model, the cumulative distribution function is distributed as a uniform distribution over the interval (0, 1). The method is first introduced for the continuous case and then for the discrete case by smoothing the cumulative distribution function. Some asymptotical properties of the method are obtained by developing an alternative to Hellys theorems. Finally, the performance of the method is evaluated by simulation, showing a good behavior.In the last years, many articles have been written about Bayesian model selection. In this article, a different and easier method is proposed and analyzed. The key idea of this method is based on the well-known property that, under the true model, the cumulative distribution function is distributed as a uniform distribution over the interval (0, 1). The method is first introduced for the continuous case and then for the discrete case by smoothing the cumulative distribution function. Some asymptotical properties of the method are obtained by developing an alternative to Hellys theorems. Finally, the performance of the method is evaluated by simulation, showing a good behavior.

Reconciling Classical and Prior Predictive P-Values in the Two-Sided Location Parameter Testing Problem

Abstract Micheas and Dey (2003) reconciled classical and Bayesian p-values in the one-sided location parameter testing problem. In this article, the classical p-value is reconciled with the prior predictive p-value, for the two-sided location parameter testing problem, proving that the classical p-value coincides with the infimum of prior predictive p-values when the prior ranges in different classes of priors.

Asymptotic behaviour of the predictive density in the exchangeable case

Abstract Consider the exchangeable case, in which ( x 1 , …, x n ) is an observation of the n -variate random variable ( X 1 , …, X n ) with density ∫ Θ П n i=1 f(x i ¦θ) d Q(θ) , where f(x¦θ) is a known model and Q is an unknown mixing distribution. The usual predictive density obtained from a prior p ( θ ) and the observation ( x 1 , …, x n ) is considered as a possible estimator for the one-variate marginal density f Q (y) = ∫ Θ f(y¦θ) d Q(θ) and its asymptotic behaviour, under P Q (the underlying probability model for X 1 , …, X n , …), is studied. We show, under mild conditions, that the predictive density is an asymptotically unbiased estimator of f Q ( y ). We also prove that it is consistent in mean squared error if and only if Q is a degenerate distribution. Finally, we consider a pooled version of this estimator and prove its L 1 -consistency (which allows to consider it as a global density estimator) and asymptotic normality.