Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of random variables within some tractable family of graphical models. In this paper, we focus our attention on the class of planar Ising models, for which inference is tractable using techniques of statistical physics [Kac and Ward; Kasteleyn]. Based on these techniques and recent methods for planarity testing and planar embedding [Chrobak and Payne], we propose a simple greedy algorithm for learning the best planar Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data). Given the set of all pairwise correlations among variables, we select a planar graph and optimal planar Ising model defined on this graph to best approximate that set of correlations. We demonstrate our method in some simulations and for the application of modeling senate voting records.

We introduce block-tree graphs as a framework for deriving efficient algorithms on graphical models. We define block-tree graphs as a tree-structured graph where each node is a cluster of nodes such that the clusters in the graph are disjoint. This differs from junction-trees, where two clusters connected by an edge always have at least one common node. When compared to junction-trees, we show that constructing block-tree graphs is faster, and finding optimal block-tree graphs has a much smaller search space. Applying our block-tree graph framework to graphical models, we show that, for some graphs, e.g., grid graphs, using block-tree graphs for inference is computationally more efficient than using junction-trees. For graphical models with boundary conditions, the block-tree graph framework transforms the boundary valued problem into an initial value problem. For Gaussian graphical models, the block-tree graph framework leads to a linear state-space representation. Since exact inference in graphical models can be computationally intractable, we propose to use spanning block-trees to derive approximate inference algorithms. Experimental results show the improved performance in using spanning block-trees versus using spanning trees for approximate estimation over Gaussian graphical models.

Spontaneous brain activity, as observed in functional neuroimaging, has been shown to display reproducible structure that expresses brain architecture and carries markers of brain pathologies. An important view of modern neuroscience is that such large-scale structure of coherent activity reflects modularity properties of brain connectivity graphs. However, to date, there has been no demonstration that the limited and noisy data available in spontaneous activity observations could be used to learn full-brain probabilistic models that generalize to new data. Learning such models entails two main challenges: i) modeling full brain connectivity is a difficult estimation problem that faces the curse of dimensionality and ii) variability between subjects, coupled with the variability of functional signals between experimental runs, makes the use of multiple datasets challenging. We describe subject-level brain functional connectivity structure as a multivariate Gaussian process and introduce a new strategy to estimate it from group data, by imposing a common structure on the graphical model in the population. We show that individual models learned from functional Magnetic Resonance Imaging (fMRI) data using this population prior generalize better to unseen data than models based on alternative regularization schemes. To our knowledge, this is the first report of a cross-validated model of spontaneous brain activity. Finally, we use the estimated graphical model to explore the large-scale characteristics of functional architecture and show for the first time that known cognitive networks appear as the integrated communities of functional connectivity graph.

Missing data estimation is an important challenge with high-dimensional data arranged in the form of a matrix. Typically this data matrix is transposable, meaning that either the rows, columns or both can be treated as features. To model transposable data, we present a modification of the matrix-variate normal, the mean-restricted matrix-variate normal, in which the rows and columns each have a separate mean vector and covariance matrix. By placing additive penalties on the inverse covariance matrices of the rows and columns, these so-called transposable regularized covariance models allow for maximum likelihood estimation of the mean and nonsingular covariance matrices. Using these models, we formulate EM-type algorithms for missing data imputation in both the multivariate and transposable frameworks. We present theoretical results exploiting the structure of our transposable models that allow these models and imputation methods to be applied to high-dimensional data. Simulations and results on microarray data and the Netflix data show that these imputation techniques often outperform existing methods and offer a greater degree of flexibility.

In a dynamic social or biological environment, the interactions between the actors can undergo large and systematic changes. In this paper we propose a model-based approach to analyze what we will refer to as the dynamic tomography of such time-evolving networks. Our approach offers an intuitive but powerful tool to infer the semantic underpinnings of each actor, such as its social roles or biological functions, underlying the observed network topologies. Our model builds on earlier work on a mixed membership stochastic blockmodel for static networks, and the state-space model for tracking object trajectory. It overcomes a major limitation of many current network inference techniques, which assume that each actor plays a unique and invariant role that accounts for all its interactions with other actors; instead, our method models the role of each actor as a time-evolving mixed membership vector that allows actors to behave differently over time and carry out different roles/functions when interacting with different peers, which is closer to reality. We present an efficient algorithm for approximate inference and learning using our model; and we applied our model to analyze a social network between monks (i.e., the Sampson's network), a dynamic email communication network between the Enron employees, and a rewiring gene interaction network of fruit fly collected during its full life cycle. In all cases, our model reveals interesting patterns of the dynamic roles of the actors.

The Bradley-Terry model is a popular approach to describe probabilities of the possible outcomes when elements of a set are repeatedly compared with one another in pairs. It has found many applications including animal behaviour, chess ranking and multiclass classification. Numerous extensions of the basic model have also been proposed in the literature including models with ties, multiple comparisons, group comparisons and random graphs. From a computational point of view, Hunter (2004) has proposed efficient iterative MM (minorization-maximization) algorithms to perform maximum likelihood estimation for these generalized Bradley-Terry models whereas Bayesian inference is typically performed using MCMC (Markov chain Monte Carlo) algorithms based on tailored Metropolis-Hastings (M-H) proposals. We show here that these MM\ algorithms can be reinterpreted as special instances of Expectation-Maximization (EM) algorithms associated to suitable sets of latent variables and propose some original extensions. These latent variables allow us to derive simple Gibbs samplers for Bayesian inference. We demonstrate experimentally the efficiency of these algorithms on a variety of applications.

In this paper, a new reinforcement learning approach is proposed which is based on a powerful concept named Active Learning Method (ALM) in modeling. ALM expresses any multi-input-single-output system as a fuzzy combination of some single-input-singleoutput systems. The proposed method is an actor-critic system similar to Generalized Approximate Reasoning based Intelligent Control (GARIC) structure to adapt the ALM by delayed reinforcement signals. Our system uses Temporal Difference (TD) learning to model the behavior of useful actions of a control system. The goodness of an action is modeled on Reward- Penalty-Plane. IDS planes will be updated according to this plane. It is shown that the system can learn with a predefined fuzzy system or without it (through random actions).

This article introduces a "pseudo classical" notion of modelling non-separability. This form of non-separability can be viewed as lying between separability and quantum-like non-separability. Non-separability is formalized in terms of the non-factorizabilty of the underlying joint probability distribution. A decision criterium for determining the non-factorizability of the joint distribution is related to determining the rank of a matrix as well as another approach based on the chi-square-goodness-of-fit test. This pseudo-classical notion of non-separability is discussed in terms of quantum games and concept combinations in human cognition.

Reasoning about Commonsense knowledge poses many problems that traditional logical inference doesn't handle well. Among these is cross-domain inference: how to draw on multiple independently produced knowledge bases. Since knowledge bases may not have the same vocabulary, level of detail, or accuracy, that inference should be "scruffy." The AnalogySpace technique showed that a factored inference approach is useful for approximate reasoning over noisy knowledge bases like ConceptNet. A straightforward extension of factored inference to multiple datasets, called Blending, has seen productive use for commonsense reasoning. We show that Blending is a kind of Collective Matrix Factorization (CMF): the factorization spreads out the prediction loss between each dataset. We then show that blending additional data causes the singular vectors to rotate between the two domains, which enables cross-domain inference. We show, in a simplified example, that the maximum interaction occurs when the magnitudes (as defined by the largest singular values) of the two matrices are equal, confirming previous empirical conclusions. Finally, we describe and mathematically justify Bridge Blending, which facilitates inference between datasets by specifically adding knowledge that "bridges" between the two, in terms of CMF.

The contribution of this paper is twofold. First, we present a new method for collective intention recognition based on mainstream philosophical accounts. Second, we extend our previous Elder Care system with collective intention recognition ability for assisting a couple of elderly people. The previous system was just capable of individual intention recognition, and so it has now been enabled to deal with situations where the elders intend to do things together.