In nonlinear latent variable models or dynamic models, if we consider the latent variables as confounders (common causes), the noise dependencies imply further relations between the observed variables. Such models are then closely related to causal discovery in the presence of nonlinear confounders, which is a challenging problem. However, generally in such models the observation noise is assumed to be independent across data dimensions, and consequently the noise dependencies are ignored. In this paper we focus on the Gaussian process latent variable model (GPLVM), from which we develop an extended model called invariant GPLVM (IGPLVM), which can adapt to arbitrary noise covariances. With the Gaussian process prior put on a particular transformation of the latent nonlinear functions, instead of the original ones, the algorithm for IGPLVM involves almost the same computational loads as that for the original GPLVM. Besides its potential application in causal discovery, IGPLVM has the advantage that its estimated latent nonlinear manifold is invariant to any nonsingular linear transformation of the data. Experimental results on both synthetic and realworld data show its encouraging performance in nonlinear manifold learning and causal discovery.

We consider learning continuous probabilistic graphical models in the face of missing data. For non-Gaussian models, learning the parameters and structure of such models depends on our ability to perform efficient inference, and can be prohibitive even for relatively modest domains. Recently, we introduced the Copula Bayesian Network (CBN) density model - a flexible framework that captures complex high-dimensional dependency structures while offering direct control over the univariate marginals, leading to improved generalization. In this work we show that the CBN model also offers significant computational advantages when training data is partially observed. Concretely, we leverage on the specialized form of the model to derive a computationally amenable learning objective that is a lower bound on the log-likelihood function. Importantly, our energy-like bound circumvents the need for costly inference of an auxiliary distribution, thus facilitating practical learning of highdimensional densities. We demonstrate the effectiveness of our approach for learning the structure and parameters of a CBN model for two reallife continuous domains.

Graphical models are widely used in scienti fic and engineering research to represent conditional independence structures between random variables. In many controlled experiments, environmental changes or external stimuli can often alter the conditional dependence between the random variables, and potentially produce significant structural changes in the corresponding graphical models. Therefore, it is of great importance to be able to detect such structural changes from data, so as to gain novel insights into where and how the structural changes take place and help the system adapt to the new environment. Here we report an effective learning strategy to extract structural changes in Gaussian graphical model using l1-regularization based convex optimization. We discuss the properties of the problem formulation and introduce an efficient implementation by the block coordinate descent algorithm. We demonstrate the principle of the approach on a numerical simulation experiment, and we then apply the algorithm to the modeling of gene regulatory networks under different conditions and obtain promising yet biologically plausible results.

Bayesian structure learning is the NP-hard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian structure learning under graph theoretic restrictions on the super-structure. The super-structure (a concept introduced by Perrier, Imoto, and Miyano, JMLR 2008) is an undirected graph that contains as subgraphs the skeletons of solution networks. Our results apply to several variants of score-based Bayesian structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth and in linear time if in addition the super-structure has bounded maximum degree. We complement this with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but we aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).

We propose a new method for estimating the intrinsic dimension of a dataset by applying the principle of regularized maximum likelihood to the distances between close neighbors. We propose a regularization scheme which is motivated by divergence minimization principles. We derive the estimator by a Poisson process approximation, argue about its convergence properties and apply it to a number of simulated and real datasets. We also show it has the best overall performance compared with two other intrinsic dimension estimators.

In recent years, a rich variety of shrinkage priors have been proposed that have great promise in addressing massive regression problems. In general, these new priors can be expressed as scale mixtures of normals, but have more complex forms and better properties than traditional Cauchy and double exponential priors. We first propose a new class of normal scale mixtures through a novel generalized beta distribution that encompasses many interesting priors as special cases. This encompassing framework should prove useful in comparing competing priors, considering properties and revealing close connections. We then develop a class of variational Bayes approximations through the new hierarchy presented that will scale more efficiently to the types of truly massive data sets that are now encountered routinely.

Motivated by applications in large-scale knowledge base construction, we study the problem of scaling up a sophisticated statistical inference framework called Markov Logic Networks (MLNs). Our approach, Felix, uses the idea of Lagrangian relaxation from mathematical programming to decompose a program into smaller tasks while preserving the joint-inference property of the original MLN. The advantage is that we can use highly scalable specialized algorithms for common tasks such as classification and coreference. We propose an architecture to support Lagrangian relaxation in an RDBMS which we show enables scalable joint inference for MLNs. We empirically validate that Felix is significantly more scalable and efficient than prior approaches to MLN inference by constructing a knowledge base from 1.8M documents as part of the TAC challenge. We show that Felix scales and achieves state-of-the-art quality numbers. In contrast, prior approaches do not scale even to a subset of the corpus that is three orders of magnitude smaller.

The decentralized particle filter (DPF) was proposed recently to increase the level of parallelism of particle filtering. Given a decomposition of the state space into two nested sets of variables, the DPF uses a particle filter to sample the first set and then conditions on this sample to generate a set of samples for the second set of variables. The DPF can be understood as a variant of the popular Rao-Blackwellized particle filter (RBPF), where the second step is carried out using Monte Carlo approximations instead of analytical inference. As a result, the range of applications of the DPF is broader than the one for the RBPF. In this paper, we improve the DPF in two ways. First, we derive a Monte Carlo approximation of the optimal proposal distribution and, consequently, design and implement a more efficient look-ahead DPF. Although the decentralized filters were initially designed to capitalize on parallel implementation, we show that the look-ahead DPF can outperform the standard particle filter even on a single machine. Second, we propose the use of bandit algorithms to automatically configure the state space decomposition of the DPF.

Sparsity-promoting priors have become increasingly popular over recent years due to an increased number of regression and classification applications involving a large number of predictors. In time series applications where observations are collected over time, it is often unrealistic to assume that the underlying sparsity pattern is fixed. We propose here an original class of flexible Bayesian linear models for dynamic sparsity modelling. The proposed class of models expands upon the existing Bayesian literature on sparse regression using generalized multivariate hyperbolic distributions. The properties of the models are explored through both analytic results and simulation studies. We demonstrate the model on a financial application where it is shown that it accurately represents the patterns seen in the analysis of stock and derivative data, and is able to detect major events by filtering an artificial portfolio of assets.

In this letter we borrow from the inference techniques developed for unbounded state-cardinality (nonparametric) variants of the HMM and use them to develop a tuning-parameter free, black-box inference procedure for Explicit-state-duration hidden Markov models (EDHMM). EDHMMs are HMMs that have latent states consisting of both discrete state-indicator and discrete state-duration random variables. In contrast to the implicit geometric state duration distribution possessed by the standard HMM, EDHMMs allow the direct parameterisation and estimation of per-state duration distributions. As most duration distributions are defined over the positive integers, truncation or other approximations are usually required to perform EDHMM inference.