In large-scale applications of undirected graphical models, such as social networks and biological networks, similar patterns occur frequently and give rise to similar parameters. In this situation, it is beneficial to group the parameters for more efficient learning. We show that even when the grouping is unknown, we can infer these parameter groups during learning via a Bayesian approach. We impose a Dirichlet process prior on the parameters. Posterior inference usually involves calculating intractable terms, and we propose two approximation algorithms, namely a Metropolis-Hastings algorithm with auxiliary variables and a Gibbs sampling algorithm with "stripped" Beta approximation (Gibbs SBA). Simulations show that both algorithms outperform conventional maximum likelihood estimation (MLE). Gibbs SBA's performance is close to Gibbs sampling with exact likelihood calculation. Models learned with Gibbs SBA also generalize better than the models learned by MLE on real-world Senate voting data.

Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatiallyvarying or non-uniform blur operator. Using ideas from Bayesian inference and convex analysis, this paper derives a simple non-uniform blind deblurring algorithm with a spatially-adaptive image penalty. Through an implicit normalization process, this penalty automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. Remaining regions with modest blur and revealing edges therefore dominate on average without explicitly incorporating structureselection heuristics. The algorithm can be implemented using an optimization strategy that is virtually tuning-parameter free and simpler than existing methods, and likely can be applied in other settings such as dictionary learning. Detailed theoretical analysis and empirical comparisons on real images serve as validation.

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be inaccurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other timevarying copula methods.

We give a polynomial-time algorithm for provably learning the structure and parameters of bipartite noisy-or Bayesian networks of binary variables where the top layer is completely hidden. Unsupervised learning of these models is a form of discrete factor analysis, enabling the discovery of hidden variables and their causal relationships with observed data. We obtain an efficient learning algorithm for a family of Bayesian networks that we call quartet-learnable. For each latent variable, the existence of a singly-coupled quartet allows us to uniquely identify and learn all parameters involving that latent variable. We give a proof of the polynomial sample complexity of our learning algorithm, and experimentally compare it to variational EM.

When a probabilistic model and its prior are given, Bayesian learning offers inference with automatic parameter tuning. However, Bayesian learning is often obstructed by computational difficulty: the rigorous Bayesian learning is intractable in many models, and its variational Bayesian (VB) approximation is prone to suffer from local minima. In this paper, we overcome this difficulty for low-rank subspace clustering (LRSC) by providing an exact global solver and its efficient approximation. LRSC extracts a low-dimensional structure of data by embedding samples into the union of low-dimensional subspaces, and its variational Bayesian variant has shown good performance. We first prove a key property that the VB-LRSC model is highly redundant. Thanks to this property, the optimization problem of VB-LRSC can be separated into small subproblems, each of which has only a small number of unknown variables. Our exact global solver relies on another key property that the stationary condition of each subproblem consists of a set of polynomial equations, which is solvable with the homotopy method. For further computational efficiency, we also propose an efficient approximate variant, of which the stationary condition can be written as a polynomial equation with a single variable. Experimental results show the usefulness of our approach.

We propose a novel class of algorithms for low rank matrix completion. Our approach builds on novel penalty functions on the singular values of the low rank matrix. By exploiting a mixture model representation of this penalty, we show that a suitably chosen set of latent variables enables to derive an Expectation-Maximization algorithm to obtain a Maximum A Posteriori estimate of the completed low rank matrix. The resulting algorithm is an iterative soft-thresholded algorithm which iteratively adapts the shrinkage coefficients associated to the singular values. The algorithm is simple to implement and can scale to large matrices. We provide numerical comparisons between our approach and recent alternatives showing the interest of the proposed approach for low rank matrix completion.

We study the problem of reconstructing low-rank matrices from their noisy observations. We formulate the problem in the Bayesian framework, which allows us to exploit structural properties of matrices in addition to low-rankedness, such as sparsity. We propose an efficient approximate message passing algorithm, derived from the belief propagation algorithm, to perform the Bayesian inference for matrix reconstruction. We have also successfully applied the proposed algorithm to a clustering problem, by reformulating it as a low-rank matrix reconstruction problem with an additional structural property. Numerical experiments show that the proposed algorithm outperforms Lloyd's K-means algorithm.

Inference in general Ising models is difficult, due to high treewidth making treebased algorithms intractable. Moreover, when interactionsarestrong,Gibbssampling may take exponential time to converge to the stationary distribution. We present an algorithm to project Ising model parameters onto aparametersetthat is guaranteed to be fast mixing, under several divergences. We find that Gibbs sampling using the projected parameters is more accurate than with the original parameters when interaction strengths are strong and when limited time is available for sampling.

The Bayesian online change point detection (BOCPD) algorithm provides an efficient way to do exact inference when the parameters of an underlying model may suddenly change over time. BOCPD requires computation of the underlying model's posterior predictives, which can only be computed online in O(1) time and memory for exponential family models. We develop variational approximations to the posterior on change point times (formulated as run lengths) for efficient inference when the underlying model is not in the exponential family, and does not have tractable posterior predictive distributions. In doing so, we develop improvements to online variational inference. We apply our methodology to a tracking problem using radar data with a signal-to-noise feature that is Rice distributed. We also develop a variational method for inferring the parameters of the (non-exponential family) Rice distribution.

The framework makes streaming updates to the estimated posterior according to a user-specified approximation batch primitive. We demonstrate the usefulness of our framework, with variational Bayes (VB) as the primitive, by fitting the latent Dirichlet allocation model to two large-scale document collections. We demonstrate the advantages of our algorithm over stochastic variational inference (SVI) by comparing the two after a single pass through a known amount of data--a case where SVI may be applied--and in the streaming setting, where SVI does not apply.