In models that define probabilities via energies, maximum likelihood learning typically involves using Markov Chain Monte Carlo to sample from the model's distribution. If the Markov chain is started at the data distribution, learning often works well even if the chain is only run for a few time steps [3]. But if the data distribution contains modes separated by regions of very low density, brief MCMC will not ensure that different modes have the correct relative energies because it cannot move particles from one mode to another. We show how to improve brief MCMC by allowing long-range moves that are suggested by the data distribution. If the model is approximately correct, these long-range moves have a reasonable acceptance rate.

In applying Hidden Markov Models to the analysis of massive data streams, it is often necessary to use an artificially reduced set of states; this is due in large part to the fact that the basic HMM estimation algorithms have a quadratic dependence on the size of the state set. We present algorithms that reduce this computational bottleneck to linear or near-linear time, when the states can be embedded in an underlying grid of parameters. This type of state representation arises in many domains; in particular, we show an application to traffic analysis at a high-volume Web site.

We describe a Markov chain method for sampling from the distribution of the hidden state sequence in a nonlinear dynamical system, given a sequence of observations. This method updates all states in the sequence simultaneously using an embedded Hidden Markov Model (HMM). An update begins with the creation of "pools" of candidate states at each time. We then define an embedded HMM whose states are indexes within these pools. Using a forward-backward dynamic programming algorithm, we can efficiently choose a state sequence with the appropriate probabilities from the exponentially large number of state sequences that pass through states in these pools. We illustrate the method in a simple one-dimensional example, and in an example showing how an embedded HMM can be used to in effect discretize the state space without any discretization error. We also compare the embedded HMM to a particle smoother on a more substantial problem of inferring human motion from 2D traces of markers.

We consider the question of how well a given distribution can be approximated with probabilistic graphical models. We introduce a new parameter, effective treewidth, that captures the degree of approximability as a tradeoff between the accuracy and the complexity of approximation. We present a simple approach to analyzing achievable tradeoffs that exploits the threshold behavior of monotone graph properties, and provide experimental results that support the approach.

The E-step boils down to computing probabilities of the hidden variables given the observed variables (evidence) and current set of parameters. The M-step then, given these probabilities, yields a new set of parameters guaranteed to increase the likelihood. In Bayesian networks, that will be the focus of this article, the M-step is usually relatively straightforward. A complication may arise in the E-step, when computing the probability of the hidden variables given the evidence becomes intractable. An often used approach is to replace the exact yet intractable inference in the E step with approximate inference, either through sampling or using a deterministic variational method. The use of a "mean-field" variational method in this context leads to an algorithm known as variational EM and can be given theinterpretation of minimizing a free energy with respect to both a tractable approximate distribution (approximate E-step) and the parameters (M-step) [2]. Loopy belief propagation [3] and variants thereof, such as generalized belief propagation [4] and expectation propagation [5], have become popular alternatives to the "mean-field" variational approaches, often yielding somewhat better approximations. And indeed, they can and have been applied for approximate inference in the E-step of the EM algorithm (see e.g.

In this paper we introduce a new underlying probabilistic model for principal component analysis (PCA). Our formulation interprets PCA as a particular Gaussian process prior on a mapping from a latent space to the observed data-space. We show that if the prior's covariance function constrains the mappings to be linear the model is equivalent to PCA, we then extend the model by considering less restrictive covariance functions which allow nonlinear mappings. This more general Gaussian process latent variable model (GPLVM) is then evaluated as an approach to the visualisation of high dimensional data for three different data-sets. Additionally our nonlinear algorithm can be further kernelised leading to'twin kernel PCA' in which a mapping between feature spaces occurs.

We introduce a class of nonstationary covariance functions for Gaussian process (GP) regression. Nonstationary covariance functions allow the model to adapt to functions whose smoothness varies with the inputs. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the regression function is controlled by a parameter, freeing one from fixing the differentiability in advance. In experiments, the nonstationary GP regression model performs well when the input space is two or three dimensions, outperforming a neural network model and Bayesian free-knot spline models, and competitive with a Bayesian neural network, but is outperformed in one dimension by a state-of-the-art Bayesian free-knot spline model.

We describe a nonparametric Bayesian approach to generalizing from few labeled examples, guided by a larger set of unlabeled objects and the assumption of a latent tree-structure to the domain. The tree (or a distribution over trees) may be inferred using the unlabeled data. A prior over concepts generated by a mutation process on the inferred tree(s) allows efficient computation of the optimal Bayesian classification function from the labeled examples. We test our approach on eight real-world datasets.

Recently, relevance vector machines (RVM) have been fashioned from a sparse Bayesian learning (SBL) framework to perform supervised learning using a weight prior that encourages sparsity of representation. The methodology incorporates an additional set of hyperparameters governing the prior, one for each weight, and then adopts a specific approximation to the full marginalization over all weights and hyperparameters. Despite its empirical success however, no rigorous motivation for this particular approximation is currently available. To address this issue, we demonstrate that SBL can be recast as the application of a rigorous variational approximation to the full model by expressing the prior in a dual form. This formulation obviates the necessity of assuming any hyperpriors and leads to natural, intuitive explanations of why sparsity is achieved in practice.

Approximation structure plays an important role in inference on loopy graphs. As a tractable structure, tree approximations have been utilized in the variational method of Ghahramani & Jordan (1997) and the sequential projection method of Frey et al. (2000). However, belief propagation represents each factor of the graph with a product of single-node messages. In this paper, belief propagation is extended to represent factors with tree approximations, by way of the expectation propagation framework. That is, each factor sends a "message" to all pairs of nodes in a tree structure. The result is more accurate inferences and more frequent convergence than ordinary belief propagation, at a lower cost than variational trees or double-loop algorithms.