934 research outputs found
Conditions for Posterior Contraction in the Sparse Normal Means Problem
The first Bayesian results for the sparse normal means problem were proven
for spike-and-slab priors. However, these priors are less convenient from a
computational point of view. In the meanwhile, a large number of continuous
shrinkage priors has been proposed. Many of these shrinkage priors can be
written as a scale mixture of normals, which makes them particularly easy to
implement. We propose general conditions on the prior on the local variance in
scale mixtures of normals, such that posterior contraction at the minimax rate
is assured. The conditions require tails at least as heavy as Laplace, but not
too heavy, and a large amount of mass around zero relative to the tails, more
so as the sparsity increases. These conditions give some general guidelines for
choosing a shrinkage prior for estimation under a nearly black sparsity
assumption. We verify these conditions for the class of priors considered by
Ghosh and Chakrabarti (2015), which includes the horseshoe and the
normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian
prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend
the number of shrinkage priors which are known to lead to posterior contraction
at the minimax estimation rate
Sparse Linear Identifiable Multivariate Modeling
In this paper we consider sparse and identifiable linear latent variable
(factor) and linear Bayesian network models for parsimonious analysis of
multivariate data. We propose a computationally efficient method for joint
parameter and model inference, and model comparison. It consists of a fully
Bayesian hierarchy for sparse models using slab and spike priors (two-component
delta-function and continuous mixtures), non-Gaussian latent factors and a
stochastic search over the ordering of the variables. The framework, which we
call SLIM (Sparse Linear Identifiable Multivariate modeling), is validated and
bench-marked on artificial and real biological data sets. SLIM is closest in
spirit to LiNGAM (Shimizu et al., 2006), but differs substantially in
inference, Bayesian network structure learning and model comparison.
Experimentally, SLIM performs equally well or better than LiNGAM with
comparable computational complexity. We attribute this mainly to the stochastic
search strategy used, and to parsimony (sparsity and identifiability), which is
an explicit part of the model. We propose two extensions to the basic i.i.d.
linear framework: non-linear dependence on observed variables, called SNIM
(Sparse Non-linear Identifiable Multivariate modeling) and allowing for
correlations between latent variables, called CSLIM (Correlated SLIM), for the
temporal and/or spatial data. The source code and scripts are available from
http://cogsys.imm.dtu.dk/slim/.Comment: 45 pages, 17 figure
- …