2,028 research outputs found
Efficient variational inference in large-scale Bayesian compressed sensing
We study linear models under heavy-tailed priors from a probabilistic
viewpoint. Instead of computing a single sparse most probable (MAP) solution as
in standard deterministic approaches, the focus in the Bayesian compressed
sensing framework shifts towards capturing the full posterior distribution on
the latent variables, which allows quantifying the estimation uncertainty and
learning model parameters using maximum likelihood. The exact posterior
distribution under the sparse linear model is intractable and we concentrate on
variational Bayesian techniques to approximate it. Repeatedly computing
Gaussian variances turns out to be a key requisite and constitutes the main
computational bottleneck in applying variational techniques in large-scale
problems. We leverage on the recently proposed Perturb-and-MAP algorithm for
drawing exact samples from Gaussian Markov random fields (GMRF). The main
technical contribution of our paper is to show that estimating Gaussian
variances using a relatively small number of such efficiently drawn random
samples is much more effective than alternative general-purpose variance
estimation techniques. By reducing the problem of variance estimation to
standard optimization primitives, the resulting variational algorithms are
fully scalable and parallelizable, allowing Bayesian computations in extremely
large-scale problems with the same memory and time complexity requirements as
conventional point estimation techniques. We illustrate these ideas with
experiments in image deblurring.Comment: 8 pages, 3 figures, appears in Proc. IEEE Workshop on Information
Theory in Computer Vision and Pattern Recognition (in conjunction with
ICCV-11), Barcelona, Spain, Nov. 201
Sparse Probit Linear Mixed Model
Linear Mixed Models (LMMs) are important tools in statistical genetics. When
used for feature selection, they allow to find a sparse set of genetic traits
that best predict a continuous phenotype of interest, while simultaneously
correcting for various confounding factors such as age, ethnicity and
population structure. Formulated as models for linear regression, LMMs have
been restricted to continuous phenotypes. We introduce the Sparse Probit Linear
Mixed Model (Probit-LMM), where we generalize the LMM modeling paradigm to
binary phenotypes. As a technical challenge, the model no longer possesses a
closed-form likelihood function. In this paper, we present a scalable
approximate inference algorithm that lets us fit the model to high-dimensional
data sets. We show on three real-world examples from different domains that in
the setup of binary labels, our algorithm leads to better prediction accuracies
and also selects features which show less correlation with the confounding
factors.Comment: Published version, 21 pages, 6 figure
Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors
Sparsity has become a key concept for solving of high-dimensional inverse
problems using variational regularization techniques. Recently, using similar
sparsity-constraints in the Bayesian framework for inverse problems by encoding
them in the prior distribution has attracted attention. Important questions
about the relation between regularization theory and Bayesian inference still
need to be addressed when using sparsity promoting inversion. A practical
obstacle for these examinations is the lack of fast posterior sampling
algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full
range of Bayesian inference methods requires being able to draw samples from
the posterior probability distribution in a fast and efficient way. This is
usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this
article, we develop and examine a new implementation of a single component
Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that
the efficiency of our Gibbs sampler increases when the level of sparsity or the
dimension of the unknowns is increased. This property is contrary to the
properties of the most commonly applied Metropolis-Hastings (MH) sampling
schemes: We demonstrate that the efficiency of MH schemes for L1-type priors
dramatically decreases when the level of sparsity or the dimension of the
unknowns is increased. Practically, Bayesian inversion for L1-type priors using
MH samplers is not feasible at all. As this is commonly believed to be an
intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also
challenges common beliefs about the applicability of sample based Bayesian
inference.Comment: 33 pages, 14 figure
Maximum-a-posteriori estimation with Bayesian confidence regions
Solutions to inverse problems that are ill-conditioned or ill-posed may have
significant intrinsic uncertainty. Unfortunately, analysing and quantifying
this uncertainty is very challenging, particularly in high-dimensional
problems. As a result, while most modern mathematical imaging methods produce
impressive point estimation results, they are generally unable to quantify the
uncertainty in the solutions delivered. This paper presents a new general
methodology for approximating Bayesian high-posterior-density credibility
regions in inverse problems that are convex and potentially very
high-dimensional. The approximations are derived by using recent concentration
of measure results related to information theory for log-concave random
vectors. A remarkable property of the approximations is that they can be
computed very efficiently, even in large-scale problems, by using standard
convex optimisation techniques. In particular, they are available as a
by-product in problems solved by maximum-a-posteriori estimation. The
approximations also have favourable theoretical properties, namely they
outer-bound the true high-posterior-density credibility regions, and they are
stable with respect to model dimension. The proposed methodology is illustrated
on two high-dimensional imaging inverse problems related to tomographic
reconstruction and sparse deconvolution, where the approximations are used to
perform Bayesian hypothesis tests and explore the uncertainty about the
solutions, and where proximal Markov chain Monte Carlo algorithms are used as
benchmark to compute exact credible regions and measure the approximation
error
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