5,794 research outputs found
Bayesian adaptation
In the need for low assumption inferential methods in infinite-dimensional
settings, Bayesian adaptive estimation via a prior distribution that does not
depend on the regularity of the function to be estimated nor on the sample size
is valuable. We elucidate relationships among the main approaches followed to
design priors for minimax-optimal rate-adaptive estimation meanwhile shedding
light on the underlying ideas.Comment: 20 pages, Propositions 3 and 5 adde
Nonparametric Bayesian methods for one-dimensional diffusion models
In this paper we review recently developed methods for nonparametric Bayesian
inference for one-dimensional diffusion models. We discuss different possible
prior distributions, computational issues, and asymptotic results
Bayes and empirical Bayes: do they merge?
Bayesian inference is attractive for its coherence and good frequentist
properties. However, it is a common experience that eliciting a honest prior
may be difficult and, in practice, people often take an {\em empirical Bayes}
approach, plugging empirical estimates of the prior hyperparameters into the
posterior distribution. Even if not rigorously justified, the underlying idea
is that, when the sample size is large, empirical Bayes leads to "similar"
inferential answers. Yet, precise mathematical results seem to be missing. In
this work, we give a more rigorous justification in terms of merging of Bayes
and empirical Bayes posterior distributions. We consider two notions of
merging: Bayesian weak merging and frequentist merging in total variation.
Since weak merging is related to consistency, we provide sufficient conditions
for consistency of empirical Bayes posteriors. Also, we show that, under
regularity conditions, the empirical Bayes procedure asymptotically selects the
value of the hyperparameter for which the prior mostly favors the "truth".
Examples include empirical Bayes density estimation with Dirichlet process
mixtures.Comment: 27 page
A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing
In the spirit of modeling inference for microarrays as multiple testing for
sparse mixtures, we present a similar approach to a simplified version of
quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the
number of tests usually reaches tens of thousands, the number of tests
performed in scans for QTL usually does not exceed several hundreds. However,
in typical cases, the sparsity of significant alternatives for QTL mapping
is in the same range as for microarrays. For methodological interest, as well
as some related applications, we also consider non-sparse mixtures. Using
simulations as well as theoretical observations we study false discovery rate
(FDR), power and misclassification probability for the Benjamini-Hochberg (BH)
procedure and its modifications, as well as for various parametric and
nonparametric Bayes and Parametric Empirical Bayes procedures. Our results
confirm the observation of Genovese and Wasserman (2002) that for small p the
misclassification error of BH is close to optimal in the sense of attaining the
Bayes oracle. This property is shared by some of the considered Bayes testing
rules, which in general perform better than BH for large or moderate 's.Comment: Published in at http://dx.doi.org/10.1214/193940307000000158 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
A Bernstein-von Mises theorem in the nonparametric right-censoring model
In the recent Bayesian nonparametric literature, many examples have been
reported in which Bayesian estimators and posterior distributions do not
achieve the optimal convergence rate, indicating that the Bernstein-von
Mises theorem does not hold. In this article, we give a positive result in
this direction by showing that the Bernstein-von Mises theorem holds in
survival models for a large class of prior processes neutral to the right. We
also show that, for an arbitrarily given convergence rate n^{-\alpha} with
0<\alpha \leq 1/2, a prior process neutral to the right can be chosen so that
its posterior distribution achieves the convergence rate n^{-\alpha}.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000052
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