Multi-output Gaussian processes provide a convenient framework for multi-task problems. An illustrative and motivating example of a multi-task problem is multi-region electrophysiological time-series data, where experimentalists are interested in both power and phase coherence between channels. Recently, Wilson and Adams (2013) proposed the spectral mixture (SM) kernel to model the spectral density of a single task in a Gaussian process framework. In this paper, we develop a novel covariance kernel for multiple outputs, called the cross-spectral mixture (CSM) kernel. This new, flexible kernel represents both the power and phase relationship between multiple observation channels.

Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) algorithms have become increasingly popular for Bayesian inference in large-scale applications. Even though these methods have proved useful in several scenarios, their performance is often limited by their bias. In this study, we propose a novel sampling algorithm that aims to reduce the bias of SG-MCMC while keeping the variance at a reasonable level. Our approach is based on a numerical sequence acceleration method, namely the Richardson-Romberg extrapolation, which simply boils down to running almost the same SG-MCMC algorithm twice in parallel with different step sizes. We illustrate our framework on the popular Stochastic Gradient Langevin Dynamics (SGLD) algorithm and propose a novel SG-MCMC algorithm referred to as Stochastic Gradient Richardson-Romberg Langevin Dynamics (SGRRLD).

Markov random fields are a popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models.

We present a method for learning Bayesian networks from data sets containingthousands of variables without the need for structure constraints. Our approachis made of two parts. The first is a novel algorithm that effectively explores thespace of possible parent sets of a node. The second part is an improvement of an existingordering-based algorithm for structure optimization. The new algorithm provablyachieves a higher score compared to its original formulation.

The local field potential (LFP) is a source of information about the broad patterns of brain activity, and the frequencies present in these time-series measurements are often highly correlated between regions. It is believed that these regions may jointly constitute a brain state,'' relating to cognition and behavior. An infinite hidden Markov model (iHMM) is proposed to model the evolution of brain states, based on electrophysiological LFP data measured at multiple brain regions. A brain state influences the spectral content of each region in the measured LFP. A new state-dependent tensor factorization is employed across brain regions, and the spectral properties of the LFPs are characterized in terms of Gaussian processes (GPs).

Bayesian networks are a popular representation of asymmetric (for example causal) relationships between random variables. Markov random fields (MRFs) are a complementary model of symmetric relationships used in computer vision, spatial modeling, and social and gene expression networks. A chain graph model under the Lauritzen-Wermuth-Frydenberg interpretation (hereafter a chain graph model) generalizes both Bayesian networks and MRFs, and can represent asymmetric and symmetric relationships together.As in other graphical models, the set of marginals from distributions in a chain graph model induced by the presence of hidden variables forms a complex model. One recent approach to the study of marginal graphical models is to consider a well-behaved supermodel. Such a supermodel of marginals of Bayesian networks, defined only by conditional independences, and termed the ordinary Markov model, was studied at length in (Evans and Richardson, 2014).In this paper, we show that special mixed graphs which we call segregated graphs can be associated, via a Markov property, with supermodels of a marginal of chain graphs defined only by conditional independences.

We propose the infinite factorial dynamic model (iFDM), a general Bayesian nonparametric model for source separation. Our model builds on the Markov Indian buffet process to consider a potentially unbounded number of hidden Markov chains (sources) that evolve independently according to some dynamics, in which the state space can be either discrete or continuous. For posterior inference, we develop an algorithm based on particle Gibbs with ancestor sampling that can be efficiently applied to a wide range of source separation problems. We evaluate the performance of our iFDM on four well-known applications: multitarget tracking, cocktail party, power disaggregation, and multiuser detection. Our experimental results show that our approach for source separation does not only outperform previous approaches, but it can also handle problems that were computationally intractable for existing approaches.

We present an algorithm for finding a chordal Markov network that maximizes any given decomposable scoring function. The algorithm is based on a recursive characterization of clique trees, and it runs in O(4 n) time for n vertices. On an eight-vertex benchmark instance, our implementation turns out to be about ten million times faster than a recently proposed, constraint satisfaction based algorithm (Corander et al., NIPS 2013). Within a few hours, it is able to solve instances up to 18 vertices, and beyond if we restrict the maximum clique size. We also study the performance of a recent integer linear programming algorithm (Bartlett and Cussens, UAI 2013).

Bayesian learning is built on an assumption that the model space contains a true reflection of the data generating mechanism. This assumption is problematic, particularly in complex data environments. Here we present a Bayesian nonparametric approach to learning that makes use of statistical models, but does not assume that the model is true. Our approach has provably better properties than using a parametric model and admits a Monte Carlo sampling scheme that can afford massive scalability on modern computer architectures. The model-based aspect of learning is particularly attractive for regularizing nonparametric inference when the sample size is small, and also for correcting approximate approaches such as variational Bayes (VB).

A key problem in structured output prediction is enabling direct optimization of the task reward function that matters for test evaluation. This paper presents a simple and computationally efficient method that incorporates task reward into maximum likelihood training. We establish a connection between maximum likelihood and regularized expected reward, showing that they are approximately equivalent in the vicinity of the optimal solution. Then we show how maximum likelihood can be generalized by optimizing the conditional probability of auxiliary outputs that are sampled proportional to their exponentiated scaled rewards. We apply this framework to optimize edit distance in the output space, by sampling from edited targets.