. 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 ....

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 continuous--time Markov process. The purpose of this paper is (i) to extend this result to a large class of f--divergences, including the total variation distance, the Hellinger distance, and not only the Kullback--Leibler 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 discrete--time. Keywords : nonlinear filtering, stability, relative entropy, Kullback--Leibler information, Hellinger...

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.

It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent

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

and posterior asymptotics

The Bayesian approach to analyzing semi-parametric models are gaining popularity in practice. For the Cox proportional hazard model, it has been shown recently that the posterior is consistent and leads to asymptotically accurate confidence intervals under a Lévy process prior on the cumulative hazard rate. The explicit expression of the posterior distribution together with independent increment structure of Lévy process play a key role in the development. However, except for one-dimensional linear regression with an unknown error distribution and binary response regression with unknown link function, even consistency of Bayesian procedures has not been studied for a general prior distribution. We consider consistency of Bayesian inference for several semi-parametric models including multiple linear regression with an unknown error distribution, exponential frailty model, generalized linear model with unknown link function, Cox proportional hazard model where the baseline hazard function is unknown, accelerated failure time models

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under misspecification

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