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Consistency issues in Bayesian Nonparametrics
 In Asymptotics, Nonparametrics and Time Series: A Tribute
, 1998
"... this paper we are mainly concerned with consistency of the posterior. Informally, the posterior is said to be consistent at a true value ` 0 if the following holds: Suppose X 1 ; X 2 ; : : : ; Xn indeed arise from P `0 , then the posterior converges to the degenerate probability ffi `0 . Alternative ..."
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this paper we are mainly concerned with consistency of the posterior. Informally, the posterior is said to be consistent at a true value ` 0 if the following holds: Suppose X 1 ; X 2 ; : : : ; Xn indeed arise from P `0 , then the posterior converges to the degenerate probability ffi `0 . Alternatively with P `0 probability 1, the posterior probability of any neighborhood U of ` 0 converges to 1. Why would a Bayesian be interested in consistency? Think of an experiment in which an experimenter generates observations from a known (to the experimenter) distribution. The observations are presented to a Bayesian. It would be embarrassing if, even with large data, the Bayesian fails to come close to finding the mechanism used by the experimenter. Consistency can be thought of as a validation of the Bayesian method. It can also be interpreted as requiring that the data, at least eventually, overrides the prior opinion. Alternatively two Bayesians, with two different priors, presented with the same data eventually agree. A result of this kind relating "merging of opinions" and posterior consistency is discussed in Diaconis and Freedman [86a]. In fact, Diaconis and Freedman [86a] (henceforth abbreviated as DF) and the ensuing discussions contain a wealth of material pertaining to posterior consistency. An early result in posterior consistency is due to Doob [48], who showed that posterior consistency obtains on a set of prior measure 1. This result does not settle the question of consistency for a particular ` 0 of interest. In smooth finite dimensional problems, different methods show (for example Berk [66]) that consistency obtains at all parameter points. Freedman [63] exhibits a prior and points of inconsistency for the infinite cell multinomial. He also showed that this p...
Sensitivity of the fractional Bayes factor to prior distributions
, 2000
"... The authors derive a measure of the sensitivity of the fractional Bayes factor, an index which is used to compare models when the priors for their respective parameters are improper, or when there is concern about robustness of the prior specification. They prove that in a large class of problems, t ..."
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The authors derive a measure of the sensitivity of the fractional Bayes factor, an index which is used to compare models when the priors for their respective parameters are improper, or when there is concern about robustness of the prior specification. They prove that in a large class of problems, this measure is a decreasing function of the fraction of the sample used to update the prior distribution before the models are compared. R ESUM E Les auteurs proposent une mesure de la sensibilite du facteur de Bayes fractionnaire, un indice de comparaison de modeles employe lorsque l'on s'inquiete de la robustesse des lois a priori sur les parametres ou que cellesci sont impropres. Ils demontrent que dans beaucoup de situations, cette mesure decrot comme fonction de la fraction de l'echantillon utilisee pour mettre a jour les lois a priori avant de comparer les modeles. 1. INTRODUCTION Suppose we are comparing two models, M 1 and M 2 , and let f i (x  # i )and# 0 i be respectively ...
Sensitivity measures of the fractional Bayes factor
, 1996
"... . Bayesian model comparison typically requires calculation of the Bayes factor. In recent years, several alternative Bayes factors have been introduced to address the problem of sensitivity to prior assumptions. Among these alternatives, the fractional Bayes factor makes an important contribution, o ..."
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. Bayesian model comparison typically requires calculation of the Bayes factor. In recent years, several alternative Bayes factors have been introduced to address the problem of sensitivity to prior assumptions. Among these alternatives, the fractional Bayes factor makes an important contribution, on the grounds of consistency, robustness and coherence. Sensitivity of the fractional Bayes factor is easy to assess when the prior distributions are proper. On the other hand, when the priors are improper, most methods lead to trivial answers. In this paper we derive a measure of the sensitivity of the fractional Bayes factor with respect to improper priors, and we illustrate a possible use of this measure for the selection of the fraction of the data to be used for training. 1. Introduction Suppose we are comparing two models, M 1 and M 2 , and let f i (x j ` i ) and \Pi 0 i be respectively the distribution of the data and the prior distribution of the parameters ` i under model M i ....
Stability and Approximation of Nonlinear Filters: an Information Theoretic Approach
, 2000
"... It has recently been proved by Clark, Ocone and Coumarbatch that the relative entropy (or Kullback Leibler information distance) between two nonlinear filters with different initial conditions is a supermartingale, hence its expectation can only decrease with time. This result was obtained for a v ..."
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It has recently been proved by Clark, Ocone and Coumarbatch that the relative entropy (or Kullback Leibler information distance) between two nonlinear filters with different initial conditions is a supermartingale, hence its expectation can only decrease with time. This result was obtained for a very general model, where the unknown state and observation processes form jointly a continuoustime Markov process. The purpose of this paper is (i) to extend this result to a large class of fdivergences, including the total variation distance, the Hellinger distance, and not only the KullbackLeibler information distance, and (ii) to consider not only robustness w.r.t. the initial condition of the filter, but also w.r.t. perturbation of the state generator. On the other hand, the model considered here is much less general, and consists of a diffusion process observed in discretetime. Keywords : nonlinear filtering, stability, relative entropy, KullbackLeibler information, Hellinger...
Proportional Mean Regression Models for Censored Data
"... A novel semiparametric regression model for censored data is proposed as an alternative to the widely used proportional hazards survival model. The proposed regression model for censored data turns out to be flexible and practically meaningful. Features include physical interpretation of the regress ..."
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A novel semiparametric regression model for censored data is proposed as an alternative to the widely used proportional hazards survival model. The proposed regression model for censored data turns out to be flexible and practically meaningful. Features include physical interpretation of the regression coefficients through the mean response time instead of the hazard functions, and a rigorous proof of consistency of the posterior distribution. It is shown that the regression model obtained by a mixture of parametric families, has a proportional mean structure (as in an accelerated failure time models). The statistical inference is based on a nonparametric Bayesian approach that uses a Dirichlet process prior for the mixing distribution. Consistency of the posterior distribution of the regression parameters in the Euclidean metric is established. Finite sample parameter estimates along with associated measure of uncertainties can be computed by a MCMC method. Simulation studies are presented to provide empirical validation of the new method. Some real data examples are provided to show the easy applicability of the proposed method.
Convergence of Posterior Distribution in the Mixture of Regressions
, 2006
"... Mixture models provide a method of modeling a complex probability distribution in terms of simpler structures. In particular, the method of mixture of regressions has received considerable attention due to its modeling flexibility and availability of convenient computational algorithms. While the th ..."
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Mixture models provide a method of modeling a complex probability distribution in terms of simpler structures. In particular, the method of mixture of regressions has received considerable attention due to its modeling flexibility and availability of convenient computational algorithms. While the theoretical justification has been successfully worked out from the frequentist point of view, its Bayesian counterpart has not been fully investigated. This paper aims to contribute to theoretical justification for the mixtures of regression model from the Bayesian perspective. In particular, we establish strong consistency of posterior distribution and determine how fast posterior distribution converges to the true value of the parameter in the context of mixture of binary regressions, Poisson regressions and Gaussian regressions.
Consistency of Bayesian Procedures for Variable Selection
, 2008
"... It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent ..."
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It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent
Consistent semiparametric Bayesian inference about a location parameter
, 1997
"... We consider the problem of Bayesian inference about the centre of symmetry of a symmetric density on the real line based on independent identically distributed observations. A result of Diaconis and Freedman shows that the posterior distribution of the location parameter may be inconsistent if (symm ..."
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We consider the problem of Bayesian inference about the centre of symmetry of a symmetric density on the real line based on independent identically distributed observations. A result of Diaconis and Freedman shows that the posterior distribution of the location parameter may be inconsistent if (symmetrized) Dirichlet process prior is used for the unknown distribution function. We choose a symmetrized Polya tree prior for the unknown density and independently choose according to a continuous and positive prior density on the real line. Suppose that the parameters of Polya tree depend only on the level m of the tree and the common values rm’s are such that ∑∞ m=1 r−1=2 m ¡∞. Then it is shown that for a large class of true symmetric densities, including the trimodal distribution of Diaconis and Freedman, the marginal posterior