Inference In Graphical Models Research Articles

Chordal Graphs to Identify Graphical Model Solutions of Maximum of Entropy Under Constraints on Marginals

We consider the problem of specifying the joint distribution of a collection of variables with maximum entropy when a set of marginals are fixed. One can easily derive that the structure of the solution joint distribution is that of a graphical model. The potential functions are then marginals at some power. We address the following question, Under which conditions on the set of constraints is it possible to fully identify the canonical exponents in the maximum entropy solution as functions of the problem structure? Literature related to this topic is somewhat scattered in disciplines such as statistical mechanics, information theory, graph theory, and inference in graphical models. In this article we gather and link results from these different fields. From this, we show that for a particular class of constraints set on marginals, the chordal maximal coherent sets of constraints, it is possible to derive the canonical exponents of the graphical model solution of the maximum entropy problem as the numbers of occurrences of separators in an associated join tree. Conversely, we present sufficient conditions to ensure that a graphical model is a solution of a maximum entropy problem.

Compressive neural activity detection with fMR images using Graphical Model Inference

In this paper, a neural activity detection framework using fMRI data is proposed. The framework consists of three stages: Prediction: PREDICTING regions of interest associated with an extraordinary amount of neural activities through Temporal Clustering Analysis (TCA). MODELLING: Categorising fMRI signals related to neural activities into event prototypes through Linear Predictive Coding (LPC). Determining the types of neural activities in terms of activation, deactivation and normality through graphical model based Bayesian inference. The experiment results demonstrate the advantages of this approach in terms of computational cost and robustness against artefacts.

A PENALTY-LOGIC SIMPLE-TRANSITION MODEL FOR STRUCTURED SEQUENCES

We study the problem of learning to infer hidden-state sequences of processes whose states and observations are propositionally or relationally factored. Unfortunately, standard exact inference techniques such as Viterbi and graphical model inference exhibit exponential complexity for these processes. The main motivation behind our work is to identify a restricted space of models, which facilitate efficient inference, yet are expressive enough to remain useful in many applications. In particular, we present the penalty-logic simple-transition model, which utilizes a very simple-transition structure where the transition cost between any two states is constant. While not appropriate for all complex processes, we argue that it is often rich enough in many applications of interest, and when it is applicable there can be inference and learning advantages compared to more general models. In particular, we show that sequential inference for this model, that is, finding a minimum-cost state sequence, efficiently reduces to a single-state minimization (SSM) problem. We then show how to define atemporal-cost models in terms of penalty logic, or weighted logical constraints, and how to use this representation for practically efficient SSM computation. We present a method for learning the weights of our model from labeled training data based on Perceptron updates. Finally, we give experiments in both propositional and relational video-interpretation domains showing advantages compared to more general models.

Graph Rigidity, Cyclic Belief Propagation, and Point Pattern Matching

A recent paper [1] proposed a provably optimal polynomial time method for performing near-isometric point pattern matching by means of exact probabilistic inference in a chordal graphical model. Its fundamental result is that the chordal graph in question is shown to be globally rigid, implying that exact inference provides the same matching solution as exact inference in a complete graphical model. This implies that the algorithm is optimal when there is no noise in the point patterns. In this paper, we present a new graph that is also globally rigid but has an advantage over the graph proposed in [1]: Its maximal clique size is smaller, rendering inference significantly more efficient. However, this graph is not chordal, and thus, standard Junction Tree algorithms cannot be directly applied. Nevertheless, we show that loopy belief propagation in such a graph converges to the optimal solution. This allows us to retain the optimality guarantee in the noiseless case, while substantially reducing both memory requirements and processing time. Our experimental results show that the accuracy of the proposed solution is indistinguishable from that in [1] when there is noise in the point patterns.

Efficient belief propagation for higher-order cliques using linear constraint nodes

Belief propagation over pairwise-connected Markov random fields has become a widely used approach, and has been successfully applied to several important computer vision problems. However, pairwise interactions are often insufficient to capture the full statistics of the problem. Higher-order interactions are sometimes required. Unfortunately, the complexity of belief propagation is exponential in the size of the largest clique. In this paper, we introduce a new technique to compute belief propagation messages in time linear with respect to clique size for a large class of potential functions over real-valued variables. We discuss how this technique can be generalized to still wider classes of potential functions at varying levels of efficiency. Also, we develop a form of nonparametric belief representation specifically designed to address issues common to networks with higher-order cliques and also to the use of guaranteed-convergent forms of belief propagation. To illustrate these techniques, we perform efficient inference in graphical models where the spatial prior of natural images is captured by 2 × 2 cliques. This approach shows significant improvement over the commonly used pairwise-connected models, and may benefit a variety of applications using belief propagation to infer images or range images, including stereo, shape-from-shading, image-based rendering, segmentation, and matting.

Graphical Model Inference in Optimal Control of Stochastic Multi-Agent Systems

In this article we consider the issue of optimal control in collaborative multi-agent systems with stochastic dynamics. The agents have a joint task in which they have to reach a number of target states. The dynamics of the agents contains additive control and additive noise, and the autonomous part factorizes over the agents. Full observation of the global state is assumed. The goal is to minimize the accumulated joint cost, which consists of integrated instantaneous costs and a joint end cost. The joint end cost expresses the joint task of the agents. The instantaneous costs are quadratic in the control and factorize over the agents. The optimal control is given as a weighted linear combination of single-agent to single-target controls. The single-agent to single-target controls are expressed in terms of diffusion processes. These controls, when not closed form expressions, are formulated in terms of path integrals, which are calculated approximately by Metropolis-Hastings sampling. The weights in the control are interpreted as marginals of a joint distribution over agent to target assignments. The structure of the latter is represented by a graphical model, and the marginals are obtained by graphical model inference. Exact inference of the graphical model will break down in large systems, and so approximate inference methods are needed. We use naive mean field approximation and belief propagation to approximate the optimal control in systems with linear dynamics. We compare the approximate inference methods with the exact solution, and we show that they can accurately compute the optimal control. Finally, we demonstrate the control method in multi-agent systems with nonlinear dynamics consisting of up to 80 agents that have to reach an equal number of target states.

Graphical model inference in optimal control of stochastic multi-agent systems

In this article we consider the issue of optimal control in collaborative multi-agent systems with stochastic dynamics. The agents have a joint task in which they have to reach a number of target s...

Inference in Graphical Gaussian Models with Edge and Vertex Symmetries with thegRcPackage forR

In this paper we present the R package gRc for statistical inference in graphical Gaussian models in which symmetry restrictions have been imposed on the concentration or partial correlation matrix. The models are represented by coloured graphs where parameters associated with edges or vertices of same colour are restricted to being identical. We describe algorithms for maximum likelihood estimation and discuss model selection issues. The paper illustrates the practical use of the gRc package.

Walk-Sums and Belief Propagation in Gaussian Graphical Models

We present a new framework based on walks in a graph for analysis and inference in Gaussian graphical models. The key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph. The weight of each walk is given by a product of edgewise partial correlation coefficients. This representation holds for a large class of Gaussian graphical models which we call walk-summable. We give a precise characterization of this class of models, and relate it to other classes including diagonally dominant, attractive, non-frustrated, and pairwise-normalizable. We provide a walk-sum interpretation of Gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles. The walk-sum perspective leads to a better understanding of Gaussian belief propagation and to stronger results for its convergence in loopy graphs.

A dynamic generating graphical model for point-sets matching

This paper presents a new dynamic generating graphical model for point-sets matching. The existing algorithms on graphical models are quite robust to noise but are susceptible to the effects of outliers. We investigated the influences of separators on point-sets matching in inference of graphical models theoretically and found that the separators that consist of outliers will interfere with message-passing, which will directly lead to failure of the existing methods. Because of this, in order to minimize the outliers in the separators, we propose a new algorithm for generating a graphical model and a corresponding Junction Tree for point-sets matching. A bi-mapping algorithm is also introduced to solve the problem of multi-mapping caused by outliers. Experiments were carried out on both synthetical data and real-world data with point-sets extracted by the Harris corner detector. The results show that the proposed algorithm is significantly more stable and possesses higher accuracy of point-sets matching, which can overcome the limitation of the sensitivity of outliers in the existing graphical models.

Using Complexity for the Estimation of Bayesian Networks

Statistical inference of graphical models has become an important tool in the reconstruction of biological networks of the type which model, for example, gene regulatory interactions. In particular, the construction of a score-based Bayesian posterior density over the space of models provides an intuitive and computationally feasible method of assessing model uncertainty and of assigning statistical confidence to structural features. One problem which frequently occurs with this approach is the tendency to overestimate the degree of model complexity. Spurious graphical features obtained in this way may affect the inference in unpredictable ways, even when using scoring techniques, such as the Bayesian Information Criterion (BIC), that are specifically designed to compensate for overfitting. In this article we propose a simple adjustment to a BIC-based scoring procedure. The method proceeds in two steps. In the first step we derive an independent estimate of the parametric complexity of the model. In the second we modify the BIC score so that the mean parametric complexity of the posterior density is equal to the estimated value. The method is applied to a set of test networks, and to a collection of genes from the yeast genome known to possess regulatory relationships. A Bayesian network model with binary responses is employed. In the examples considered, we find that the number of spurious graph edges inferred is reduced, while the effect on the identification of true edges is minimal.

Approximating the problem, not the solution: An alternative view of point set matching

This work discusses the issue of approximation in point set matching. In general, one may have two classes of approximations when tackling a matching problem: (1) an algorithmic approximation which consists in using suboptimal procedures to infer the assignment, and (2), a representational approximation which involves a simplified and suboptimal model for the original data. Matching techniques have typically relied on the first approach by retaining the complete model and using suboptimal techniques to solve it. In this paper, we show how a technique based on using exact inference in simple Graphical Models, an instance of the second class, can significantly outperform instances of techniques from the first class. We experimentally compare this method with well-known Spectral and Relaxation methods, which are exemplars of the first class. We have performed experiments with synthetic and real-world data sets which reveal significant performance improvement in a wide operating range.

A comparison of algorithms for inference and learning in probabilistic graphical models

Research into methods for reasoning under uncertainty is currently one of the most exciting areas of artificial intelligence, largely because it has recently become possible to record, store, and process large amounts of data. While impressive achievements have been made in pattern classification problems such as handwritten character recognition, face detection, speaker identification, and prediction of gene function, it is even more exciting that researchers are on the verge of introducing systems that can perform large-scale combinatorial analyses of data, decomposing the data into interacting components. For example, computational methods for automatic scene analysis are now emerging in the computer vision community. These methods decompose an input image into its constituent objects, lighting conditions, motion patterns, etc. Two of the main challenges are finding effective representations and models in specific applications and finding efficient algorithms for inference and learning in these models. In this paper, we advocate the use of graph-based probability models and their associated inference and learning algorithms. We review exact techniques and various approximate, computationally efficient techniques, including iterated conditional modes, the expectation maximization (EM) algorithm, Gibbs sampling, the mean field method, variational techniques, structured variational techniques and the sum-product algorithm ("loopy" belief propagation). We describe how each technique can be applied in a vision model of multiple, occluding objects and contrast the behaviors and performances of the techniques using a unifying cost function, free energy.

Implementing belief propagation in neural circuits

There is growing evidence that neural circuits may employ statistical algorithms for inference and learning. Many such algorithms can be derived from independence diagrams ( graphical models) showing causal relationships between random variables. A general algorithm for inference in graphical models is belief propagation, where nodes in a graphical model determine values for random variables by combining observed values with messages passed between neighboring nodes. We propose that small groups of synaptic connections between neurons in cortex correspond to causal dependencies in an underlying graphical model. Our results suggest a new probabilistic framework for computation in the neocortex.

Tropical geometry of statistical models

This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. Here, we address the question of how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. The Newton polytope of a statistical model plays a key role. Our results are applied to the hidden Markov model and the general Markov model on a binary tree.

Linear response algorithms for approximate inference in graphical models.

Belief propagation (BP) on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. However, it does not prescribe a way to compute joint distributions over pairs of distant nodes in the graph. In this article, we propose two new algorithms for approximating these pairwise probabilities, based on the linear response theorem. The first is a propagation algorithm that is shown to converge if BP converges to a stable fixed point. The second algorithm is based on matrix inversion. Applying these ideas to gaussian random fields, we derive a propagation algorithm for computing the inverse of a matrix.

Extended message passing algorithm for inference in loopy Gaussian graphical models

We consider message passing for probabilistic inference in undirected Gaussian graphical models. We show that for singly connected graphs, message passing yields an algorithm that is equivalent to the application of Gaussian elimination to the solution of a particular system of equations. This relation provides a natural way of extending message passing to arbitrary graphs with loops by first studying the operations required by Gaussian elimination. We thus obtain a finite time convergent algorithm that solves the inference problem exactly and whose complexity grows gradually with the “distance” of the graph to a tree. This algorithm can be implemented in a distributed fashion at nodes through message passing, as for example in sensor networks.

Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non‐decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY‐conjugate prior (Diaconis & Ylvisaker, 1979) for non‐decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution (Dawid & Lauritzen, 1993). In particular, if G is an incomplete prime graph it constitutes a non‐trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non‐decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non‐decomposable zero‐pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.

Bayesian inference for graphical factor analysis models

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years.

Improving the Mean-Field Approximation in Belief Networks Using Bahadur's Reparameterisation of the Multivariate Binary Distribution

We develop a new extension to the Mean-Field approximation for inference in graphical models which has advantages over other approximation schemes which have been proposed. The method is economical in its use of variational parameters and the approximating conditional distribution can be specified with direct reference to the dependence structure of the variables in the graphical model. We apply the method to sigmoid belief networks.