Sampling inference methods are computationally difficult to scale for many models in part because global dependencies can reduce opportunities for parallel computation. Without strict conditional independence structure among variables, standard Gibbs sampling theory requires sample updates to be performed sequentially, even if dependence between most variables is not strong. Empirical work has shown that some models can be sampled effectively by going Hogwild'' and simply running Gibbs updates in parallel with only periodic global communication, but the successes and limitations of such a strategy are not well understood. As a step towards such an understanding, we study the Hogwild Gibbs sampling strategy in the context of Gaussian distributions. We develop a framework which provides convergence conditions and error bounds along with simple proofs and connections to methods in numerical linear algebra.

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parametrization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increase quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.

EDML is a recently proposed algorithm for learning parameters in Bayesian networks. It was originally derived in terms of approximate inference on a meta-network, which underlies the Bayesian approach to parameter estimation. While this initial derivation helped discover EDML in the first place and provided a concrete context for identifying some of its properties (e.g., in contrast to EM), the formal setting was somewhat tedious in the number of concepts it drew on. In this paper, we propose a greatly simplified perspective on EDML, which casts it as a general approach to continuous optimization. The new perspective has several advantages.

We address the problem of learning a sparse Bayesian network structure for continuous variables in a high-dimensional space. The constraint that the estimated Bayesian network structure must be a directed acyclic graph (DAG) makes the problem challenging because of the huge search space of network structures. Most previous methods were based on a two-stage approach that prunes the search space in the first stage and then searches for a network structure that satisfies the DAG constraint in the second stage. Although this approach is effective in a low-dimensional setting, it is difficult to ensure that the correct network structure is not pruned in the first stage in a high-dimensional setting. In this paper, we propose a single-stage method, called A* lasso, that recovers the optimal sparse Bayesian network structure by solving a single optimization problem with A* search algorithm that uses lasso in its scoring system. Our approach substantially improves the computational efficiency of the well-known exact methods based on dynamic programming. We also present a heuristic scheme that further improves the efficiency of A* lasso without significantly compromising the quality of solutions and demonstrate this on benchmark Bayesian networks and real data.

Reliance on computationally expensive algorithms for inference has been limiting the use of Bayesian nonparametric models in large scale applications. To tackle this problem, we propose a Bayesian learning algorithm for DP mixture models. Instead of following the conventional paradigm -- random initialization plus iterative update, we take an progressive approach. Starting with a given prior, our method recursively transforms it into an approximate posterior through sequential variational approximation. In this process, new components will be incorporated on the fly when needed. The algorithm can reliably estimate a DP mixture model in one pass, making it particularly suited for applications with massive data. Experiments on both synthetic data and real datasets demonstrate remarkable improvement on efficiency -- orders of magnitude speed-up compared to the state-of-the-art.

We describe a class of algorithms for amortized inference in Bayesian networks. In this setting, we invest computation upfront to support rapid online inference for a wide range of queries. Our approach is based on learning an inverse factorization of a model's joint distribution: a factorization that turns observations into root nodes. Our algorithms accumulate information to estimate the local conditional distributions that constitute such a factorization. These stochastic inverses can be used to invert each of the computation steps leading to an observation, sampling backwards in order to quickly find a likely explanation. We show that estimated inverses converge asymptotically in number of (prior or posterior) training samples. To make use of inverses before convergence, we describe the Inverse MCMC algorithm, which uses stochastic inverses to make block proposals for a Metropolis-Hastings sampler. We explore the efficiency of this sampler for a variety of parameter regimes and Bayes nets.

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'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.

The receptive field (RF) of a sensory neuron describes how the neuron integrates sensory stimuli over time and space. In typical experiments with naturalistic or flickering spatiotemporal stimuli, RFs are very high-dimensional, due to the large number of coefficients needed to specify an integration profile across time and space. Estimating these coefficients from small amounts of data poses a variety of challenging statistical and computational problems. Here we address these challenges by developing Bayesian reduced rank regression methods for RF estimation. This corresponds to modeling the RF as a sum of several space-time separable (i.e., rank-1) filters, which proves accurate even for neurons with strongly oriented space-time RFs.

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once an approximation of the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically.

Coverage functions are an important class of discrete functions that capture laws of diminishing returns. In this paper, we propose a new problem of learning time-varying coverage functions which arise naturally from applications in social network analysis, machine learning, and algorithmic game theory. We develop a novel parametrization of the time-varying coverage function by illustrating the connections with counting processes. We present an efficient algorithm to learn the parameters by maximum likelihood estimation, and provide a rigorous theoretic analysis of its sample complexity. Empirical experiments from information diffusion in social network analysis demonstrate that with few assumptions about the underlying diffusion process, our method performs significantly better than existing approaches on both synthetic and real world data.