Bayesian compressive sensing (CS) is considered for signals and images that are sparse in a wavelet basis. The statistical structure of the wavelet coefficients is exploited explicitly in the proposed model, and therefore this framework goes beyond simply assuming that the data are compressible in a wavelet basis. The structure exploited within the wavelet coefficients is consistent with that used in waveletbased compression algorithms. A hierarchical Bayesian model is constituted, with efficient inference via Markov chain Monte Carlo (MCMC) sampling. The algorithm is fully developed and demonstrated using several natural images, with performance comparisons to many state-of-the-art compressive-sensing inversion algorithms.

We present a nonparametric hierarchical Bayesian model of document collections that decouples sparsity and smoothness in the component distributions (i.e., the “topics”). In the sparse topic model (sparseTM), each topic is represented by a bank of selector variables that determine which terms appear in the topic. Thus each topic is associated with a subset of the vocabulary, and topic smoothness is modeled on this subset. We develop an efficient Gibbs sampler for the sparseTM that includes a general-purpose method for sampling from a Dirichlet mixture with a combinatorial number of components. We demonstrate the sparseTM on four real-world datasets. Compared to traditional approaches, the empirical results will show that sparseTMs give better predictive performance with simpler inferred models. 1

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We propose a factor-graph-based approach to joint channel-estimationand-decoding of bit-interleaved coded orthogonal frequency division multiplexing (BICM-OFDM). In contrast to existing designs, ours is capable of exploiting not only sparsity in sampled channel taps but also clustering among the large taps, behaviors which are known to manifest at larger communication bandwidths. In order to exploit these channel-tap structures, we adopt a two-state Gaussian mixture prior in conjunction with a Markov model on the hidden state. For loopy belief propagation, we exploit a “generalized approximate message passing ” algorithm recently developed in the context of compressed sensing, and show that it can be successfully coupled with soft-input soft-output decoding, as well as hidden Markov inference. ForN subcarriers andM bits per subcarrier (and any channel length L < N), our scheme has a computational complexity of onlyO(N log 2N+N2 M). Numerical experiments using IEEE 802.15.4a channels show that our scheme yields BER performance within 1 dB of the known-channel bound and 4 dB better than decoupled channel-estimation-and-decoding via LASSO. 1.

Editor: A logistic stick-breaking process (LSBP) is proposed for non-parametric clustering of general spatially- or temporally-dependent data, imposing the belief that proximate data are more likely to be clustered together. The sticks in the LSBP are realized via multiple logistic regression functions, with shrinkage priors employed to favor contiguous and spatially localized segments. The LSBP is also extended for the simultaneous processing of multiple data sets, yielding a hierarchical logistic stick-breaking process (H-LSBP). The model parameters (atoms) within the H-LSBP are shared across the multiple learning tasks. Efficient variational Bayesian inference is derived, and comparisons are made to related techniques in the literature. Experimental analysis is performed for audio waveforms and images, and it is demonstrated that for segmentation applications the LSBP yields generally homogeneous segments with sharp boundaries.

Identifying co-varying causal elements in very high dimensional feature space with internal structures, e.g., a space with as many as millions of linearly ordered features, as one typically encounters in problems such as whole genome association (WGA) mapping, remains an open problem in statistical learning. We propose a block-regularized regression model for sparse variable selection in a high-dimensional space where the covariates are linearly ordered, and are possibly subject to local statistical linkages (e.g., block structures) due to spacial or temporal proximity of the features. Our goal is to identify a small subset of relevant covariates that are not merely from random positions in the ordering, but grouped as contiguous blocks from large number of ordered covariates. Following a typical linear regression framework between the features and the response, our proposed model employs a sparsity-enforcing Laplacian prior for the regression coefficients, augmented by a 1st-order Markovian process along the feature sequence that “activates” the regression coefficients in a coupled fashion. We describe a sampling-based learning algorithm and demonstrate the performance of our method on simulated and biological data for marker identification under WGA. 1

In this paper we propose a method for the automatic decipherment of lost languages. Given a non-parallel corpus in a known related language, our model produces both alphabetic mappings and translations of words into their corresponding cognates. We employ a non-parametric Bayesian framework to simultaneously capture both low-level character mappings and highlevel morphemic correspondences. This formulation enables us to encode some of the linguistic intuitions that have guided human decipherers. When applied to the ancient Semitic language Ugaritic, the model correctly maps 29 of 30 letters to their Hebrew counterparts, and deduces the correct Hebrew cognate for 60 % of the Ugaritic words which have cognates in Hebrew. 1

Description Spike and slab for prediction and variable selection in linear regression models. Uses a generalized elastic net for variable selection.

Choosing the subset of covariates to use in regression or generalized linear models is a ubiquitous problem. The Bayesian paradigm can easily deal with this problem of model uncertainty by considering models corresponding to all possible subsets of the covariates, where the posterior distribution over models is used to select models or combine them in Bayesian model averaging. Although conceptually straightforward, it is often difficult to implement in practice, since either the number of covariates is too large, or calculations cannot be done analytically, or both. For orthogonal designs with the appropriate choice of prior, the posterior probability of any model can be calculated without having to enumerate the entire model space. In this article we propose a novel method, which augments the observed non-orthogonal design by new rows to obtain a design matrix with orthogonal columns. We show that our data augmentation approach keeps the original posterior distribution of interest unaltered, and develop methods to construct Rao-Blackwellized estimates of several quantities of interest, including posterior model probabilities, which may not be available from an ordinary Gibbs sampler. The method can be used for BMA in linear regression with Cauchy or other heavy tailed priors that may be represented as a scale mixture of normals, as well as binary regression. We provide simulated and real examples to illustrate the methodology. Supplemental materials for the manuscript are available online.

Abstract Weighted generalized ridge regression offers unique advantages in correlated highdimensional problems. Such estimators can be efficiently computed using Bayesian spike and slab models and are effective for prediction. For sparse variable selection, a generalization of the elastic net can be used in tandem with these Bayesian estimates. In this article, we describe the R-software package spikeslab for implementing this new spike and slab prediction and variable selection methodology. The expression spike and slab, originally coined by Mitchell and Beauchamp (1988), refers to a type of prior used for the regression coefficients in linear regression models (see also Lempers (1971)). In Mitchell and Beauchamp (1988), this prior assumed that the regression coefficients were mutually independent with a two-point mixture distribution made up of a uniform flat distribution (the slab) and a degenerate distribution at zero (the spike). In George and McCulloch (1993) a different prior for the regression coefficient was used. This involved a scale (variance) mixture of two normal distributions. In particular, the use of a normal prior was instrumental in facilitating efficient Gibbs sampling of the posterior. This made spike and slab variable selection computationally attractive and heavily popularized the method. As pointed out in Ishwaran and Rao (2005), normal-scale mixture priors, such as those used in George and McCulloch (1993), constitute a wide class of models termed spike and slab models. Spike and slab models were extended to the class of rescaled spike and slab models (Ishwaran and Rao, 2005). Rescaling was shown to induce a nonvanishing penalization effect, and when used in tandem with a continuous bimodal prior, confers useful model selection properties for the posterior mean of