Factorial Hidden Markov Models

Hidden Markov models (HMMs) have proven to be one of the most widely used tools for learning probabilistic models of time series data. In an HMM, information about the past is conveyed through a single discrete variable—the hidden state. We discuss a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. We describe an exact algorithm for inferring the posterior probabilities of the hidden state variables given the observations, and relate it to the forward–backward algorithm for HMMs and to algorithms for more general graphical models. Due to the combinatorial nature of the hidden state representation, this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or variational methods. Within the variational framework, we present a structured approximation in which the the state variables are decoupled, yielding a tractable algorithm for learning the parameters of the model. Empirical comparisons suggest that these approximations are efficient and provide accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bach's chorales and show that factorial HMMs can capture statistical structure in this data set which an unconstrained HMM cannot.     

Association Models for Web Mining

We describe how statistical association models and, specifically, graphical models, can be usefully employed to model web mining data. We describe some methodological problems related to the implementation of discrete graphical models for web mining data. In particular, we discuss model selection procedures.     

Models and Methods of Finite-Dimensional Variational Inequalities

Algorithms for solving finite-dimensional inequalities are studied. The emphasis is on numerical methods based on the optimization approach. Examples of economic equilibrium models that assist in solving variational inequalities are presented.     

Choice of Basis for Laplace Approximation

Maximum a posteriori optimization of parameters and the Laplace approximation for the marginal likelihood are both basis-dependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the 'softmax' basis.     

Approximate inference in Boltzmann machines

Inference in Boltzmann machines is NP-hard in general. As a result approximations are often necessary. We discuss first order mean field and second order Onsager truncations of the Plefka expansion of the Gibbs free energy. The Bethe free energy is introduced and rewritten as a Gibbs free energy. From there a convergent belief optimization algorithm is derived to minimize the Bethe free energy. An analytic expression for the linear response estimate of the covariances is found which is exact on Boltzmann trees. Finally, a number of theorems is proven concerning the Plefka expansion, relating the first order mean field and the second order Onsager approximation to the Bethe approximation. Experiments compare mean field approximation, Onsager approximation, belief propagation and belief optimization.     

A Constructive Graphical Model Approach for Knowledge-Based Systems: A Vehicle Monitoring Case Study

Graphical models have been widely applied to uncertain reasoning in knowledge-based systems. For many of the problems tackled, a single graphical model is constructed before individual cases are presented and the model is used to reason about each new case. In this work, we consider a class of problems whose solution requires inference over a very large number of models that are impractical to construct a priori. We conduct a case study in the domain of vehicle monitoring and then generalize the approach taken. We show that the previously held negative belief on the applicability of graphical models to such problems is unjustified. We propose a set of techniques based on domain decomposition, model separation, model approximation, model compilation, and re-analysis to meet the computational challenges imposed by the combinatorial explosion. Experimental results on vehicle monitoring demonstrated good performance at near-real-time.     

A Bayesian method for learning belief networks that contain hidden variables

This paper presents a Bayesian method for computing the probability of a Bayesian belief-network structure from a database. In particular, the paper focuses on computing the probability of a belief-network structure that contains a hidden (latent) variable. A hidden variable represents a postulated entity that has not been directly measured. After reviewing related techniques, which previously were reported, this paper presents a new, more efficient method for handling hidden variables in belief networks.     

Variational Bayesian methods for spatial data analysis

With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. However, fitting spatial models often involves expensive matrix decompositions, whose computational complexity increases in cubic order with the number of spatial locations. This situation is aggravated in Bayesian settings where such computations are required once at every iteration of the Markov chain Monte Carlo (MCMC) algorithms. In this paper, we describe the use of Variational Bayesian (VB) methods as an alternative to MCMC to approximate the posterior distributions of complex spatial models. Variational methods, which have been used extensively in Bayesian machine learning for several years, provide a lower bound on the marginal likelihood, which can be computed efficiently. We provide results for the variational updates in several models especially emphasizing their use in multivariate spatial analysis. We demonstrate estimation and model comparisons from VB methods by using simulated data as well as environmental data sets and compare them with inference from MCMC.     

Population Markov Chain Monte Carlo

Stochastic search algorithms inspired by physical and biological systems are applied to the problem of learning directed graphical probability models in the presence of missing observations and hidden variables. For this class of problems, deterministic search algorithms tend to halt at local optima, requiring random restarts to obtain solutions of acceptable quality. We compare three stochastic search algorithms: a Metropolis-Hastings Sampler (MHS), an Evolutionary Algorithm (EA), and a new hybrid algorithm called Population Markov Chain Monte Carlo, or popMCMC. PopMCMC uses statistical information from a population of MHSs to inform the proposal distributions for individual samplers in the population. Experimental results show that popMCMC and EAs learn more efficiently than the MHS with no information exchange. Populations of MCMC samplers exhibit more diversity than populations evolving according to EAs not satisfying physics-inspired local reversibility conditions.