7,877 research outputs found
On nonparametric estimation of a mixing density via the predictive recursion algorithm
Nonparametric estimation of a mixing density based on observations from the
corresponding mixture is a challenging statistical problem. This paper surveys
the literature on a fast, recursive estimator based on the predictive recursion
algorithm. After introducing the algorithm and giving a few examples, I
summarize the available asymptotic convergence theory, describe an important
semiparametric extension, and highlight two interesting applications. I
conclude with a discussion of several recent developments in this area and some
open problems.Comment: 22 pages, 5 figures. Comments welcome at
https://www.researchers.one/article/2018-12-
A nonparametric empirical Bayes framework for large-scale multiple testing
We propose a flexible and identifiable version of the two-groups model,
motivated by hierarchical Bayes considerations, that features an empirical null
and a semiparametric mixture model for the non-null cases. We use a
computationally efficient predictive recursion marginal likelihood procedure to
estimate the model parameters, even the nonparametric mixing distribution. This
leads to a nonparametric empirical Bayes testing procedure, which we call
PRtest, based on thresholding the estimated local false discovery rates.
Simulations and real-data examples demonstrate that, compared to existing
approaches, PRtest's careful handling of the non-null density can give a much
better fit in the tails of the mixture distribution which, in turn, can lead to
more realistic conclusions.Comment: 18 pages, 4 figures, 3 table
Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It
We empirically show that Bayesian inference can be inconsistent under
misspecification in simple linear regression problems, both in a model
averaging/selection and in a Bayesian ridge regression setting. We use the
standard linear model, which assumes homoskedasticity, whereas the data are
heteroskedastic, and observe that the posterior puts its mass on ever more
high-dimensional models as the sample size increases. To remedy the problem, we
equip the likelihood in Bayes' theorem with an exponent called the learning
rate, and we propose the Safe Bayesian method to learn the learning rate from
the data. SafeBayes tends to select small learning rates as soon the standard
posterior is not `cumulatively concentrated', and its results on our data are
quite encouraging.Comment: 70 pages, 20 figure
Bayesian Compressed Regression
As an alternative to variable selection or shrinkage in high dimensional
regression, we propose to randomly compress the predictors prior to analysis.
This dramatically reduces storage and computational bottlenecks, performing
well when the predictors can be projected to a low dimensional linear subspace
with minimal loss of information about the response. As opposed to existing
Bayesian dimensionality reduction approaches, the exact posterior distribution
conditional on the compressed data is available analytically, speeding up
computation by many orders of magnitude while also bypassing robustness issues
due to convergence and mixing problems with MCMC. Model averaging is used to
reduce sensitivity to the random projection matrix, while accommodating
uncertainty in the subspace dimension. Strong theoretical support is provided
for the approach by showing near parametric convergence rates for the
predictive density in the large p small n asymptotic paradigm. Practical
performance relative to competitors is illustrated in simulations and real data
applications.Comment: 29 pages, 4 figure
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