Restricted Boltzmann machines (RBMs) [36, 9, 20] have recently attracted significant interest due to their versatility in a variety of unsupervised and supervised learning tasks [35, 18, 25], and in building deep architectures [14, 31]. A RBM is a bipartite undirected model that captures the generative process in which a data vector is generated from a binary hidden vector. The bipartite architecture enables very fast data encoding and sampling-based inference; and together with recent advances in learning procedures, we can now process massive data with large models [13, 37, 2]. This paper presents our contributions in developing RBM specifications as well as learning and inference procedures for multivariate ordinal data. This extends and consolidates the reach of RBMs to a wide range of user-generated domains - social responses, recommender systems, product/paper reviews, and expert assessments of health and ecosystems indicators.

Feature selection, identifying a subset of variables that are relevant for predicting a response, is an important and challenging component of many methods in statistics and machine learning. Feature selection is especially difficult and computationally intensive when the number of variables approaches or exceeds the number of samples, as is often the case for many genomic datasets. Here, we introduce a new approach -- the Bayesian Ising Approximation (BIA) -- to rapidly calculate posterior probabilities for feature relevance in L2 penalized linear regression. In the regime where the regression problem is strongly regularized by the prior, we show that computing the marginal posterior probabilities for features is equivalent to computing the magnetizations of an Ising model. Using a mean field approximation, we show it is possible to rapidly compute the feature selection path described by the posterior probabilities as a function of the L2 penalty. We present simulations and analytical results illustrating the accuracy of the BIA on some simple regression problems. Finally, we demonstrate the applicability of the BIA to high dimensional regression by analyzing a gene expression dataset with nearly 30,000 features.

Credal networks are graph-based statistical models whose parameters take values in a set, instead of being sharply specified as in traditional statistical models (e.g., Bayesian networks). The computational complexity of inferences on such models depends on the irrelevance/independence concept adopted. In this paper, we study inferential complexity under the concepts of epistemic irrelevance and strong independence. We show that inferences under strong independence are NP-hard even in trees with binary variables except for a single ternary one. We prove that under epistemic irrelevance the polynomial-time complexity of inferences in credal trees is not likely to extend to more general models (e.g., singly connected topologies). These results clearly distinguish networks that admit efficient inferences and those where inferences are most likely hard, and settle several open questions regarding their computational complexity. We show that these results remain valid even if we disallow the use of zero probabilities. We also show that the computation of bounds on the probability of the future state in a hidden Markov model is the same whether we assume epistemic irrelevance or strong independence, and we prove a similar result for inference in naive Bayes structures. These inferential equivalences are important for practitioners, as hidden Markov models and naive Bayes structures are used in real applications of imprecise probability.

In Bayesian analysis the form of the posterior distribution is often not analytically tractable. To obtain quantities of interest under such a distribution, such as moments or marginal distributions, we typically need to use Monte Carlo methods or approximate the posterior with a more convenient distribution. A popular method of obtaining such an approximation is structured or fixed-form Variational Bayes, which works by numerically minimizing the Kullback-Leibler divergence of an approximating distribution in the exponential family to the intractable target distribution (Attias, 2000; Beal and Ghahramani, 2006; Jordan et al., 1999; Wainwright and Jordan, 2008). For certain problems, algorithms exist that can solve this optimization problem in much less time than it would take to approximate the posterior using Monte Carlo methods (see e.g.

Motivation: Algorithms that discover variables which are causally related to a target may inform the design of experiments. With observational gene expression data, many methods discover causal variables by measuring each variable's degree of statistical dependence with the target using dependence measures (DMs). However, other methods measure each variable's ability to explain the statistical dependence between the target and the remaining variables in the data using conditional dependence measures (CDMs), since this strategy is guaranteed to find the target's direct causes, direct effects, and direct causes of the direct effects in the infinite sample limit. In this paper, we design a new algorithm in order to systematically compare the relative abilities of DMs and CDMs in discovering causal variables from gene expression data. Results: The proposed algorithm using a CDM is sample efficient, since it consistently outperforms other state-of-the-art local causal discovery algorithms when samples sizes are small. However, the proposed algorithm using a CDM outperforms the proposed algorithm using a DM only when sample sizes are above several hundred. These results suggest that accurate causal discovery from gene expression data using current CDM-based algorithms requires datasets with at least several hundred samples. Availability: The proposed algorithm is freely available at https://github.com/ericstrobl/DvCD.

We introduce a logic for reasoning about evidence, that essentially views evidence as a function from prior beliefs (before making an observation) to posterior beliefs (after making the observation). We provide a sound and complete axiomatization for the logic, and consider the complexity of the decision problem. Although the reasoning in the logic is mainly propositional, we allow variables representing numbers and quantification over them. This expressive power seems necessary to capture important properties of evidence.

We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between conditioning and calibration when uncertainty is described by sets of probabilities.

It is commonly-accepted wisdom that more information is better, and that information should never be ignored. Here we argue, using both a Bayesian and a non-Bayesian analysis, that in some situations you are better off ignoring information if your uncertainty is represented by a set of probability measures. These include situations in which the information is relevant for the prediction task at hand. In the non-Bayesian analysis, we show how ignoring information avoids dilation, the phenomenon that additional pieces of information sometimes lead to an increase in uncertainty. In the Bayesian analysis, we show that for small sample sizes and certain prediction tasks, the Bayesian posterior based on a noninformative prior yields worse predictions than simply ignoring the given information.

A general notion of algebraic conditional plausibility measures is defined. Probability measures, ranking functions, possibility measures, and (under the appropriate definitions) sets of probability measures can all be viewed as defining algebraic conditional plausibility measures. It is shown that the technology of Bayesian networks can be applied to algebraic conditional plausibility measures.

In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the `best' one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw away all but the best model, we average multiple models that are in some sense close to the best one, thereby producing a weighted average of clustering results. Two (weighted) averaging approaches are considered: averaging the component membership probabilities and averaging models. In both cases, Occam's window is used to determine closeness to the best model and weights are computed within a Bayesian model averaging paradigm. In some cases, we need to merge components before averaging; we introduce a method for merging mixture components based on the adjusted Rand index. The effectiveness of our model-based clustering averaging approaches is illustrated using a family of Gaussian mixture models on real and simulated data.