7 research outputs found
Sparsity oracle inequalities for the Lasso
This paper studies oracle properties of -penalized least squares in
nonparametric regression setting with random design. We show that the penalized
least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in
terms of the number of non-zero components of the oracle vector. The results
are valid even when the dimension of the model is (much) larger than the sample
size and the regression matrix is not positive definite. They can be applied to
high-dimensional linear regression, to nonparametric adaptive regression
estimation and to the problem of aggregation of arbitrary estimators.Comment: Published at http://dx.doi.org/10.1214/07-EJS008 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Fast Exact Bayesian Inference for Sparse Signals in the Normal Sequence Model
We consider exact algorithms for Bayesian inference with model selection
priors (including spike-and-slab priors) in the sparse normal sequence model.
Because the best existing exact algorithm becomes numerically unstable for
sample sizes over n=500, there has been much attention for alternative
approaches like approximate algorithms (Gibbs sampling, variational Bayes,
etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO)
or empirical Bayesian methods. However, by introducing algorithmic ideas from
online sequential prediction, we show that exact calculations are feasible for
much larger sample sizes: for general model selection priors we reach n=25000,
and for certain spike-and-slab priors we can easily reach n=100000. We further
prove a de Finetti-like result for finite sample sizes that characterizes
exactly which model selection priors can be expressed as spike-and-slab priors.
The computational speed and numerical accuracy of the proposed methods are
demonstrated in experiments on simulated data, on a differential gene
expression data set, and to compare the effect of multiple hyper-parameter
settings in the beta-binomial prior. In our experimental evaluation we compute
guaranteed bounds on the numerical accuracy of all new algorithms, which shows
that the proposed methods are numerically reliable whereas an alternative based
on long division is not
SPADES and mixture models
This paper studies sparse density estimation via penalization
(SPADES). We focus on estimation in high-dimensional mixture models and
nonparametric adaptive density estimation. We show, respectively, that SPADES
can recover, with high probability, the unknown components of a mixture of
probability densities and that it yields minimax adaptive density estimates.
These results are based on a general sparsity oracle inequality that the SPADES
estimates satisfy. We offer a data driven method for the choice of the tuning
parameter used in the construction of SPADES. The method uses the generalized
bisection method first introduced in \citebb09. The suggested procedure
bypasses the need for a grid search and offers substantial computational
savings. We complement our theoretical results with a simulation study that
employs this method for approximations of one and two-dimensional densities
with mixtures. The numerical results strongly support our theoretical findings.Comment: Published in at http://dx.doi.org/10.1214/09-AOS790 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences
We consider full Bayesian inference in the multivariate normal mean model in
the situation that the mean vector is sparse. The prior distribution on the
vector of means is constructed hierarchically by first choosing a collection of
nonzero means and next a prior on the nonzero values. We consider the posterior
distribution in the frequentist set-up that the observations are generated
according to a fixed mean vector, and are interested in the posterior
distribution of the number of nonzero components and the contraction of the
posterior distribution to the true mean vector. We find various combinations of
priors on the number of nonzero coefficients and on these coefficients that
give desirable performance. We also find priors that give suboptimal
convergence, for instance, Gaussian priors on the nonzero coefficients. We
illustrate the results by simulations.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1029 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Spike and slab empirical Bayes sparse credible sets
In the sparse normal means model, coverage of adaptive Bayesian posterior
credible sets associated to spike and slab prior distributions is considered.
The key sparsity hyperparameter is calibrated via marginal maximum likelihood
empirical Bayes. First, adaptive posterior contraction rates are derived with
respect to --type--distances for . Next, under a type of
so-called excessive-bias conditions, credible sets are constructed that have
coverage of the true parameter at prescribed confidence level and at
the same time are of optimal diameter. We also prove that the previous
conditions cannot be significantly weakened from the minimax perspective.Comment: 45 page