In this paper we demonstrate that tempering Markov chain Monte Carlo samplers for Bayesian models by recursively subsampling observations without replacement can improve the performance of baseline samplers in terms of effective sample size per computation. We present two tempering by subsampling algorithms, subsampled parallel tempering and subsampled tempered transitions. We provide an asymptotic analysis of the computational cost of tempering by subsampling, verify that tempering by subsampling costs less than traditional tempering, and demonstrate both algorithms on Bayesian approaches to learning the mean of a high dimensional multivariate Normal and estimating Gaussian process hyperparameters.

We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-specific contexts results in the nDP mixture over content variables. We provide a Polya-urn view of the model and an efficient collapsed Gibbs inference procedure. Extensive experiments on real-world datasets demonstrate the advantage of utilizing context information via our model in both text and image domains.

Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with currently known algorithms, sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices. In this paper, we design reversible irreducible Markov chains on the space of Markov equivalent classes by proposing a perfect set of operators that determine the transitions of the Markov chain. The stationary distribution of a proposed Markov chain has a closed form and can be computed easily. Specifically, we construct a concrete perfect set of operators on sparse Markov equivalence classes by introducing appropriate conditions on each possible operator. Algorithms and their accelerated versions are provided to efficiently generate Markov chains and to explore properties of Markov equivalence classes of sparse directed acyclic graphs (DAGs) with thousands of vertices. We find experimentally that in most Markov equivalence classes of sparse DAGs, (1) most edges are directed, (2) most undirected subgraphs are small and (3) the number of these undirected subgraphs grows approximately linearly with the number of vertices. The article contains supplement arXiv:1303.0632, http://dx.doi.org/10.1214/13-AOS1125SUPP

In this paper, computer-based techniques for stylistic analysis of paintings are applied to the five panels of the 14th century Peruzzi Altarpiece by Giotto di Bondone. Features are extracted by combining a dual-tree complex wavelet transform with a hidden Markov tree (HMT) model. Hierarchical clustering is used to identify stylistic keywords in image patches, and keyword frequencies are calculated for sub-images that each contains many patches. A generative hierarchical Bayesian model learns stylistic patterns of keywords; these patterns are then used to characterize the styles of the sub-images; this in turn, permits to discriminate between paintings. Results suggest that such unsupervised probabilistic topic models can be useful to distill characteristic elements of style.

Generative Stochastic Networks (GSNs) have been recently introduced as an alternative to traditional probabilistic modeling: instead of parametrizing the data distribution directly, one parametrizes a transition operator for a Markov chain whose stationary distribution is an estimator of the data generating distribution. The result of training is therefore a machine that generates samples through this Markov chain. However, the previously introduced GSN consistency theorems suggest that in order to capture a wide class of distributions, the transition operator in general should be multimodal, something that has not been done before this paper. We introduce for the first time multimodal transition distributions for GSNs, in particular using models in the NADE family (Neural Autoregressive Density Estimator) as output distributions of the transition operator. A NADE model is related to an RBM (and can thus model multimodal distributions) but its likelihood (and likelihood gradient) can be computed easily. The parameters of the NADE are obtained as a learned function of the previous state of the learned Markov chain. Experiments clearly illustrate the advantage of such multimodal transition distributions over unimodal GSNs.

The study of networks has received increased attention recently not only from the social sciences and statistics but also from physicists, computer scientists and mathematicians. One of the principal problem in networks is community detection. Many algorithms have been proposed for community finding but most of them do not have have theoretical guarantee for sparse networks and networks close to the phase transition boundary proposed by physicists. There are some exceptions but all have some incomplete theoretical basis. Here we propose an algorithm based on the graph distance of vertices in the network. We give theoretical guarantees that our method works in identifying communities for block models and can be extended for degree-corrected block models and block models with the number of communities growing with number of vertices. Despite favorable simulation results, we are not yet able to conclude that our method is satisfactory for worst possible case. We illustrate on a network of political blogs, Facebook networks and some other networks.

Hierarchical parametric models consisting of observable and latent variables are widely used for unsupervised learning tasks. For example, a mixture model is a representative hierarchical model for clustering. From the statistical point of view, the models can be regular or singular due to the distribution of data. In the regular case, the models have the identifiability; there is one-to-one relation between a probability density function for the model expression and the parameter. The Fisher information matrix is positive definite, and the estimation accuracy of both observable and latent variables has been studied. In the singular case, on the other hand, the models are not identifiable and the Fisher matrix is not positive definite. Conventional statistical analysis based on the inverse Fisher matrix is not applicable. Recently, an algebraic geometrical analysis has been developed and is used to elucidate the Bayes estimation of observable variables. The present paper applies this analysis to latent-variable estimation and determines its theoretical performance. Our results clarify behavior of the convergence of the posterior distribution. It is found that the posterior of the observable-variable estimation can be different from the one in the latent-variable estimation. Because of the difference, the Markov chain Monte Carlo method based on the parameter and the latent variable cannot construct the desired posterior distribution.

We introduce a general framework for measuring risk in the context of Markov control processes with risk maps on general Borel spaces that generalize known concepts of risk measures in mathematical finance, operations research and behavioral economics. Within the framework, applying weighted norm spaces to incorporate also unbounded costs, we study two types of infinite-horizon risk-sensitive criteria, discounted total risk and average risk, and solve the associated optimization problems by dynamic programming. For the discounted case, we propose a new discount scheme, which is different from the conventional form but consistent with the existing literature, while for the average risk criterion, we state Lyapunov-like stability conditions that generalize known conditions for Markov chains to ensure the existence of solutions to the optimality equation.

We present a theoretical analysis of Gaussian-binary restricted Boltzmann machines (GRBMs) from the perspective of density models. The key aspect of this analysis is to show that GRBMs can be formulated as a constrained mixture of Gaussians, which gives a much better insight into the model's capabilities and limitations. We show that GRBMs are capable of learning meaningful features both in a two-dimensional blind source separation task and in modeling natural images. Further, we show that reported difficulties in training GRBMs are due to the failure of the training algorithm rather than the model itself. Based on our analysis we are able to propose several training recipes, which allowed successful and fast training in our experiments. Finally, we discuss the relationship of GRBMs to several modifications that have been proposed to improve the model.

Discovering causal relations among observed variables in a given data set is a major objective in studies of statistics and artificial intelligence. Recently, some techniques to discover a unique causal model have been explored based on non-Gaussianity of the observed data distribution. However, most of these are limited to continuous data. In this paper, we present a novel causal model for binary data and propose an efficient new approach to deriving the unique causal model governing a given binary data set under skew distributions of external binary noises. Experimental evaluation shows excellent performance for both artificial and real world data sets.