349 research outputs found
Bayes Model Selection with Path Sampling: Factor Models and Other Examples
We prove a theorem justifying the regularity conditions which are needed for
Path Sampling in Factor Models. We then show that the remaining ingredient,
namely, MCMC for calculating the integrand at each point in the path, may be
seriously flawed, leading to wrong estimates of Bayes factors. We provide a new
method of Path Sampling (with Small Change) that works much better than
standard Path Sampling in the sense of estimating the Bayes factor better and
choosing the correct model more often. When the more complex factor model is
true, PS-SC is substantially more accurate. New MCMC diagnostics is provided
for these problems in support of our conclusions and recommendations. Some of
our ideas for diagnostics and improvement in computation through small changes
should apply to other methods of computation of the Bayes factor for model
selection.Comment: Published in at http://dx.doi.org/10.1214/12-STS403 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large sample asymptotics for the two-parameter Poisson--Dirichlet process
This paper explores large sample properties of the two-parameter
Poisson--Dirichlet Process in two contexts. In a Bayesian
context of estimating an unknown probability measure, viewing this process as a
natural extension of the Dirichlet process, we explore the consistency and weak
convergence of the the two-parameter Poisson--Dirichlet posterior process. We
also establish the weak convergence of properly centered two-parameter
Poisson--Dirichlet processes for large This latter result
complements large results for the Dirichlet process and
Poisson--Dirichlet sequences, and complements a recent result on large
deviation principles for the two-parameter Poisson--Dirichlet process. A
crucial component of our results is the use of distributional identities that
may be useful in other contexts.Comment: Published in at http://dx.doi.org/10.1214/074921708000000147 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
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
Consistency of a recursive estimate of mixing distributions
Mixture models have received considerable attention recently and Newton
[Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for
estimating a mixing distribution. We prove almost sure consistency of this
recursive estimate in the weak topology under mild conditions on the family of
densities being mixed. This recursive estimate depends on the data ordering and
a permutation-invariant modification is proposed, which is an average of the
original over permutations of the data sequence. A Rao--Blackwell argument is
used to prove consistency in probability of this alternative estimate. Several
simulations are presented, comparing the finite-sample performance of the
recursive estimate and a Monte Carlo approximation to the permutation-invariant
alternative along with that of the nonparametric maximum likelihood estimate
and a nonparametric Bayes estimate.Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic Bayes-optimality under sparsity of some multiple testing procedures
Within a Bayesian decision theoretic framework we investigate some asymptotic
optimality properties of a large class of multiple testing rules. A parametric
setup is considered, in which observations come from a normal scale mixture
model and the total loss is assumed to be the sum of losses for individual
tests. Our model can be used for testing point null hypotheses, as well as to
distinguish large signals from a multitude of very small effects. A rule is
defined to be asymptotically Bayes optimal under sparsity (ABOS), if within our
chosen asymptotic framework the ratio of its Bayes risk and that of the Bayes
oracle (a rule which minimizes the Bayes risk) converges to one. Our main
interest is in the asymptotic scheme where the proportion p of "true"
alternatives converges to zero. We fully characterize the class of fixed
threshold multiple testing rules which are ABOS, and hence derive conditions
for the asymptotic optimality of rules controlling the Bayesian False Discovery
Rate (BFDR). We finally provide conditions under which the popular
Benjamini-Hochberg (BH) and Bonferroni procedures are ABOS and show that for a
wide class of sparsity levels, the threshold of the former can be approximated
by a nonrandom threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AOS869 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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