Undirected graphical models known as Markov networks are popular for a wide variety of applications ranging from statistical physics to computational biology. Traditionally, learning of the network structure has been done under the assumption of chordality which ensures that efficient scoring methods can be used. In general, non-chordal graphs have intractable normalizing constants which renders the calculation of Bayesian and other scores difficult beyond very small-scale systems. Recently, there has been a surge of interest towards the use of regularized pseudo-likelihood methods for structural learning of large-scale Markov network models, as such an approach avoids the assumption of chordality. The currently available methods typically necessitate the use of a tuning parameter to adapt the level of regularization for a particular dataset, which can be optimized for example by cross-validation. Here we introduce a Bayesian version of pseudo-likelihood scoring of Markov networks, which enables an automatic regularization through marginalization over the nuisance parameters in the model. We prove consistency of the resulting MPL estimator for the network structure via comparison with the pseudo information criterion. Identification of the MPL-optimal network on a prescanned graph space is considered with both greedy hill climbing and exact pseudo-Boolean optimization algorithms. We find that for reasonable sample sizes the hill climbing approach most often identifies networks that are at a negligible distance from the restricted global optimum. Using synthetic and existing benchmark networks, the marginal pseudo-likelihood method is shown to generally perform favorably against recent popular inference methods for Markov networks.

In this paper, the use of the Generalized Beta Mixture (GBM) and Horseshoe distributions as priors in the Bayesian Compressive Sensing framework is proposed. The distributions are considered in a two-layer hierarchical model, making the corresponding inference problem amenable to Expectation Maximization (EM). We present an explicit, algebraic EM-update rule for the models, yielding two fast and experimentally validated algorithms for signal recovery. Experimental results show that our algorithms outperform state-of-the-art methods on a wide range of sparsity levels and amplitudes in terms of reconstruction accuracy, convergence rate and sparsity. The largest improvement can be observed for sparse signals with high amplitudes.

In real world industrial applications of topic modeling, the ability to capture gigantic conceptual space by learning an ultra-high dimensional topical representation, i.e., the so-called "big model", is becoming the next desideratum after enthusiasms on "big data", especially for fine-grained downstream tasks such as online advertising, where good performances are usually achieved by regression-based predictors built on millions if not billions of input features. The conventional data-parallel approach for training gigantic topic models turns out to be rather inefficient in utilizing the power of parallelism, due to the heavy dependency on a centralized image of "model". Big model size also poses another challenge on the storage, where available model size is bounded by the smallest RAM of nodes. To address these issues, we explore another type of parallelism, namely model-parallelism, which enables training of disjoint blocks of a big topic model in parallel. By integrating data-parallelism with model-parallelism, we show that dependencies between distributed elements can be handled seamlessly, achieving not only faster convergence but also an ability to tackle significantly bigger model size. We describe an architecture for model-parallel inference of LDA, and present a variant of collapsed Gibbs sampling algorithm tailored for it. Experimental results demonstrate the ability of this system to handle topic modeling with unprecedented amount of 200 billion model variables only on a low-end cluster with very limited computational resources and bandwidth.

Identifying latent structure in large data matrices is essential for exploring biological processes. Here, we consider recovering gene co-expression networks from gene expression data, where each network encodes relationships between genes that are locally co-regulated by shared biological mechanisms. To do this, we develop a Bayesian statistical model for biclustering to infer subsets of co-regulated genes whose covariation may be observed in only a subset of the samples. Our biclustering method, BicMix, has desirable properties, including allowing overcomplete representations of the data, computational tractability, and jointly modeling unknown confounders and biological signals. Compared with related biclustering methods, BicMix recovers latent structure with higher precision across diverse simulation scenarios. Further, we develop a method to recover gene co-expression networks from the estimated sparse biclustering matrices. We apply BicMix to breast cancer gene expression data and recover a gene co-expression network that is differential across ER+ and ER- samples.

Determinantal point processes (DPPs) offer a powerful approach to modeling diversity in many applications where the goal is to select a diverse subset. We study the problem of learning the parameters (the kernel matrix) of a DPP from labeled training data. We make two contributions. First, we show how to reparameterize a DPP's kernel matrix with multiple kernel functions, thus enhancing modeling flexibility. Second, we propose a novel parameter estimation technique based on the principle of large margin separation. In contrast to the state-of-the-art method of maximum likelihood estimation, our large-margin loss function explicitly models errors in selecting the target subsets, and it can be customized to trade off different types of errors (precision vs. recall). Extensive empirical studies validate our contributions, including applications on challenging document and video summarization, where flexibility in modeling the kernel matrix and balancing different errors is indispensable.

The problem of f-divergence estimation is important in the fields of machine learning, information theory, and statistics. While several nonparametric divergence estimators exist, relatively few have known convergence properties. In particular, even for those estimators whose MSE convergence rates are known, the asymptotic distributions are unknown. We establish the asymptotic normality of a recently proposed ensemble estimator of f-divergence between two distributions from a finite number of samples. This estimator has MSE convergence rate of O(1/T), is simple to implement, and performs well in high dimensions. This theory enables us to perform divergence-based inference tasks such as testing equality of pairs of distributions based on empirical samples. We experimentally validate our theoretical results and, as an illustration, use them to empirically bound the best achievable classification error.

A determinantal point process (DPP) is a probabilistic model of set diversity compactly parameterized by a positive semi-definite kernel matrix. To fit a DPP to a given task, we would like to learn the entries of its kernel matrix by maximizing the log-likelihood of the available data. However, log-likelihood is non-convex in the entries of the kernel matrix, and this learning problem is conjectured to be NP-hard. Thus, previous work has instead focused on more restricted convex learning settings: learning only a single weight for each row of the kernel matrix, or learning weights for a linear combination of DPPs with fixed kernel matrices. In this work we propose a novel algorithm for learning the full kernel matrix. By changing the kernel parameterization from matrix entries to eigenvalues and eigenvectors, and then lower-bounding the likelihood in the manner of expectation-maximization algorithms, we obtain an effective optimization procedure. We test our method on a real-world product recommendation task, and achieve relative gains of up to 16.5% in test log-likelihood compared to the naive approach of maximizing likelihood by projected gradient ascent on the entries of the kernel matrix.

Simple approximate MAP Inference for Dirichlet processes Yordan P. Raykov, Alexis Boukouvalas, Max A. Little October 27, 2014 Abstract The Dirichlet process mixture (DPM) is a ubiquitous, flexible Bayesian nonparametric statistical model. However, full probabilistic inference in this model is analytically intractable, so that computationally intensive techniques such as Gibb's sampling are required. As a result, DPM-based methods, which have considerable potential, are restricted to applications in which computational resources and time for inference is plentiful. For example, they would not be practical for digital signal processing on embedded hardware, where computational resources are at a serious premium. Here, we develop simplified yet statistically rigorous approximate maximum a-posteriori (MAP) inference algorithms for DPMs. This algorithm is as simple asK -means clustering, performs in experiments as well as Gibb's sampling, while requiring only a fraction of the computational effort. Unlike related small variance asymptotics, our algorithm is non-degenerate and so inherits the "rich get richer" property of the Dirichlet process. It also retains a non-degenerate closed-form likelihood which enables standard tools such as cross-validation to be used. This is a well-posed approximation to the MAP solution of the probabilistic DPM model. 1 Introduction Bayesian nonparametric (BNP) models have been successfully applied on a wide range of domains but despite significant improvements in computational hardware, statistical inference in most BNP models remains infeasible in the context of large datasets. The flexibility gained by such models is paid for with severe decreases in computational efficiency, and this makes these models somewhat impractical.

Identifying small subsets of features that are relevant for prediction and/or classification tasks is a central problem in machine learning and statistics. The feature selection task is especially important, and computationally difficult, for modern datasets where the number of features can be comparable to, or even exceed, the number of samples. Here, we show that feature selection with Bayesian inference takes a universal form and reduces to calculating the magnetizations of an Ising model, under some mild conditions. Our results exploit the observation that the evidence takes a universal form for strongly-regularizing priors --- priors that have a large effect on the posterior probability even in the infinite data limit. We derive explicit expressions for feature selection for generalized linear models, a large class of statistical techniques that include linear and logistic regression. We illustrate the power of our approach by analyzing feature selection in a logistic regression-based classifier trained to distinguish between the letters B and D in the notMNIST dataset.

State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient varia-tional Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dynamical systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo.