We describe computationally efficient methods for learning mixtures in which each component is a directed acyclic graphical model (mixtures of DAGs or MDAGs). We argue that simple search-and-score algorithms are infeasible for a variety of problems, and introduce a feasible approach in which parameter and structure search is interleaved and expected data is treated as real data. Our approach can be viewed as a combination of (1) the Cheeseman--Stutz asymptotic approximation for model posterior probability and (2) the Expectation--Maximization algorithm. We evaluate our procedure for selecting among MDAGs on synthetic and real examples.

We discuss Bayesian methods for learning Bayesian networks when data sets are incomplete. In particular, we examine asymptotic approximations for the marginal likelihood of incomplete data given a Bayesian network. We consider the Laplace approximation and the less accurate but more efficient BIC/MDL approximation. We also consider approximations proposed by Draper (1993) and Cheeseman and Stutz (1995). These approximations are as efficient as BIC/MDL, but their accuracy has not been studied in any depth. We compare the accuracy of these approximations under the assumption that the Laplace approximation is the most accurate. In experiments using synthetic data generated from discrete naive-Bayes models having a hidden root node, we find that the CS measure is the most accurate.

In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional L\'{e}vy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordinators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions. We exploit these subordinators in regression problems, yielding a hierarchical model with multiple regularization parameters. We devise ECME (Expectation/Conditional Maximization Either) algorithms to simultaneously estimate regression coefficients and regularization parameters. The empirical evaluation of simulated data shows that our approach is feasible and effective in high-dimensional data analysis.

Outlier detection aims to identify unusual data instances that deviate from expected patterns. The outlier detection is particularly challenging when outliers are context dependent and when they are defined by unusual combinations of multiple outcome variable values. In this paper, we develop and study a new conditional outlier detection approach for multivariate outcome spaces that works by (1) transforming the conditional detection to the outlier detection problem in a new (unconditional) space and (2) defining outlier scores by analyzing the data in the new space. Our approach relies on the classifier chain decomposition of the multi-dimensional classification problem that lets us transform the output space into a probability vector, one probability for each dimension of the output space. Outlier scores applied to these transformed vectors are then used to detect the outliers. Experiments on multiple multi-dimensional classification problems with the different outlier injection rates show that our methodology is robust and able to successfully identify outliers when outliers are either sparse (manifested in one or very few dimensions) or dense (affecting multiple dimensions).

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application.

We give a new consistent scoring function for structure learning of Bayesian networks. In contrast to traditional approaches to score-based structure learning, such as BDeu or MDL, the complexity penalty that we propose is data-dependent and is given by the probability that a conditional independence test correctly shows that an edge cannot exist. What really distinguishes this new scoring function from earlier work is that it has the property of becoming computationally easier to maximize as the amount of data increases. We prove a polynomial sample complexity result, showing that maximizing this score is guaranteed to correctly learn a structure with no false edges and a distribution close to the generating distribution, whenever there exists a Bayesian network which is a perfect map for the data generating distribution. Although the new score can be used with any search algorithm, in our related UAI 2013 paper [BS13], we have given empirical results showing that it is particularly effective when used together with a linear programming relaxation approach to Bayesian network structure learning. The present paper contains all details of the proofs of the finite-sample complexity results in [BS13] as well as detailed explanation of the computation of the certain error probabilities called beta-values, whose precomputation and tabulation is necessary for the implementation of the algorithm in [BS13].

Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach which samples from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), especially when noise and missing data are present. In addition, existing methods cannot take into account uncertainty information of latent factors, resulting in low generalization performance. To address these issues, we present a class of probabilistic generative Tucker models for tensor decomposition and completion with structural sparsity over multilinear latent space. To exploit structural sparse modeling, we introduce two group sparsity inducing priors by hierarchial representation of Laplace and Student-t distributions, which facilitates fully posterior inference. For model learning, we derived variational Bayesian inferences over all model (hyper)parameters, and developed efficient and scalable algorithms based on multilinear operations. Our methods can automatically adapt model complexity and infer an optimal multilinear rank by the principle of maximum lower bound of model evidence. Experimental results and comparisons on synthetic, chemometrics and neuroimaging data demonstrate remarkable performance of our models for recovering ground-truth of multilinear rank and missing entries.

In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easily handles data sizes not considered thus far by alternate approaches.

We suggest a novel method of clustering and exploratory analysis of temporal event sequences data (also known as categorical time series) based on three-dimensional data grid models. A data set of temporal event sequences can be represented as a data set of three-dimensional points, each point is defined by three variables: a sequence identifier, a time value and an event value. Instantiating data grid models to the 3D-points turns the problem into 3D-coclustering. The sequences are partitioned into clusters, the time variable is discretized into intervals and the events are partitioned into clusters. The cross-product of the univariate partitions forms a multivariate partition of the representation space, i.e., a grid of cells and it also represents a nonparametric estimator of the joint distribution of the sequences, time and events dimensions. Thus, the sequences are grouped together because they have similar joint distribution of time and events, i.e., similar distribution of events along the time dimension. The best data grid is computed using a parameter-free Bayesian model selection approach. We also suggest several criteria for exploiting the resulting grid through agglomerative hierarchies, for interpreting the clusters of sequences and characterizing their components through insightful visualizations. Extensive experiments on both synthetic and real-world data sets demonstrate that data grid models are efficient, effective and discover meaningful underlying patterns of categorical time series data.