51 research outputs found
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Bayesian Estimation of the Discrepancy with Misspecified Parametric Models
We study a Bayesian model where we have made specific requests about the parameter values to be estimated. The aim is to find the parameter of a parametric family which minimizes a distance to the data generating density and then to estimate the discrepancy using nonparametric methods. We illustrate how coherent updating can proceed given that the standard Bayesian posterior from an unidentifiable model is inappropriate. Our updating is performed using Markov Chain Monte Carlo methods and in particular a novel method for dealing with intractable normalizing constants is required. Illustrations using synthetic data are provided.European Research Council (ERC) through StG "N-BNP" 306406Regione PiemonteMathematic
Discussion on article “Bayesian inference with misspecified models” by Stephen G. Walker
A reversible allelic partition process and Pitman sampling formula
We introduce a continuous-time Markov chain describing dynamic allelic
partitions which extends the branching process construction of the Pitman
sampling formula in Pitman (2006) and the birth-and-death process with
immigration studied in Karlin and McGregor (1967), in turn related to the
celebrated Ewens sampling formula. A biological basis for the scheme is
provided in terms of a population of individuals grouped into families, that
evolves according to a sequence of births, deaths and immigrations. We
investigate the asymptotic behaviour of the chain and show that, as opposed to
the birth-and-death process with immigration, this construction maintains in
the temporal limit the mutual dependence among the multiplicities. When the
death rate exceeds the birth rate, the system is shown to have reversible
distribution identified as a mixture of Pitman sampling formulae, with negative
binomial mixing distribution on the population size. The population therefore
converges to a stationary random configuration, characterised by a finite
number of families and individuals.Comment: 17 pages, to appear in ALEA , Latin American Journal of Probability
and Mathematical Statistic
Asymptotics for posterior hazards
An important issue in survival analysis is the investigation and the modeling
of hazard rates. Within a Bayesian nonparametric framework, a natural and
popular approach is to model hazard rates as kernel mixtures with respect to a
completely random measure. In this paper we provide a comprehensive analysis of
the asymptotic behavior of such models. We investigate consistency of the
posterior distribution and derive fixed sample size central limit theorems for
both linear and quadratic functionals of the posterior hazard rate. The general
results are then specialized to various specific kernels and mixing measures
yielding consistency under minimal conditions and neat central limit theorems
for the distribution of functionals.Comment: Published in at http://dx.doi.org/10.1214/08-AOS631 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Bernstein-Von Mises Theorem in Semiparametric Competing Risks Models
Semiparametric Bayesian models are nowadays a popular tool in survival analysis. An important area of research concerns the investigation of frequentist properties of these models. In this paper, a Bernstein-von Mises theorem is derived for semiparametric Bayesian models of competing risks data. The cause-specific hazard is taken as the product of the conditional probability of a failure type and the overall hazard rate. We model the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We show that the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.Bayesian nonparametrics, Bernstein-von Mises theorem, beta process, competing risks, conditional probability of a failure type, semiparametric inference.
Asymptotics for posterior hazards
An important issue in survival analysis is the investigation and the modeling of hazard rates. Within a Bayesian nonparametric framework, a natural and popular approach is to model hazard rates as kernel mixtures with respect to a completely random measure. In this paper we provide a comprehensive analysis of the asymptotic behavior of such models. We investigate consistency of the posterior distribution and derive fixed sample size central limit theorems for both linear and quadratic functionals of the posterior hazard rate. The general results are then specialized to various specific kernels and mixing measures yielding consistency under minimal conditions and neat central limit theorems for the distribution of functionals.Bayesian Consistency; Bayesian Nonparametrics; Central limit theorem; Completely random measure; Path-variance; Random hazard rate; Survival analysis
A class of neutral to the right priors induced by superposition of beta processes
A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.Bayesian nonparametrics; beta process; beta-Stacy process; completely random measures; neutral to the right priors; survival analysis
Bayesian nonparametric estimation and consistency of mixed multinomial logit choice models
This paper develops nonparametric estimation for discrete choice models based
on the mixed multinomial logit (MMNL) model. It has been shown that MMNL models
encompass all discrete choice models derived under the assumption of random
utility maximization, subject to the identification of an unknown distribution
. Noting the mixture model description of the MMNL, we employ a Bayesian
nonparametric approach, using nonparametric priors on the unknown mixing
distribution , to estimate choice probabilities. We provide an important
theoretical support for the use of the proposed methodology by investigating
consistency of the posterior distribution for a general nonparametric prior on
the mixing distribution. Consistency is defined according to an -type
distance on the space of choice probabilities and is achieved by extending to a
regression model framework a recent approach to strong consistency based on the
summability of square roots of prior probabilities. Moving to estimation,
slightly different techniques for non-panel and panel data models are
discussed. For practical implementation, we describe efficient and relatively
easy-to-use blocked Gibbs sampling procedures. These procedures are based on
approximations of the random probability measure by classes of finite
stick-breaking processes. A simulation study is also performed to investigate
the performance of the proposed methods.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ233 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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