The nonnegative Boltzmann machine (NNBM) is a recurrent neural net(cid:173) work model that can describe multimodal nonnegative data. Application of maximum likelihood estimation to this model gives a learning rule that is analogous to the binary Boltzmann machine. We examine the utility of the mean field approximation for the NNBM, and describe how Monte Carlo sampling techniques can be used to learn its parameters. Reflec(cid:173) tive slice sampling is particularly well-suited for this distribution, and can efficiently be implemented to sample the distribution. We illustrate learning of the NNBM on a transiationally invariant distribution, as well as on a generative model for images of human faces.

We present a class of approximate inference algorithms for graphical models of the QMR-DT type. We give convergence rates for these al(cid:173) gorithms and for the Jaakkola and Jordan (1999) algorithm, and verify these theoretical predictions empirically.

We consider the problem of learning a grid-based map using a robot with noisy sensors and actuators. We compare two approaches: online EM, where the map is treated as a fixed parameter, and Bayesian inference, where the map is a (matrix-valued) random variable. We show that even on a very simple example, online EM can get stuck in local minima, which causes the robot to get "lost" and the resulting map to be useless. By contrast, the Bayesian approach, by maintaining multiple hypotheses, is much more ro(cid:173) bust. We then introduce a method for approximating the Bayesian solution, called Rao-Blackwellised particle filtering.

In recent years, Bayesian networks have become highly successful tool for di(cid:173) agnosis, analysis, and decision making in real-world domains. We present an efficient algorithm for learning Bayes networks from data. In contrast to the majority of work, which typically uses hill-climbing approaches that may produce dense and causally incorrect nets, our approach yields much more compact causal networks by heeding independencies in the data. Compact causal networks facilitate fast in(cid:173) ference and are also easier to understand. We prove that under mild assumptions, our approach requires time polynomial in the size of the data and the number of nodes.

Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. While a sim(cid:173) ple variational argument shows that Bayes averaging is generaliza(cid:173) tion optimal given that the prior matches the teacher parameter distribution the situation is less clear if the teacher distribution is unknown. I define a class of averaging procedures, the temperated likelihoods, including both Bayes averaging with a uniform prior and maximum likelihood estimation as special cases. I show that Bayes is generalization optimal in this family for any teacher dis(cid:173) tribution for two learning problems that are analytically tractable: learning the mean of a Gaussian and asymptotics of smooth learn(cid:173) ers.

We estimate a statistical model of typical activities from a large set of 3D periodic human motion data by segmenting these data automatically into "cycles". Then the mean and the princi(cid:173) pal components of the cycles are computed using a new algorithm that accounts for missing information and enforces smooth tran(cid:173) sitions between cycles. The learned temporal model provides a prior probability distribution over human motions that can be used in a Bayesian framework for tracking human subjects in complex monocular video sequences and recovering their 3D motion.

Preliminary work by the authors made use of the so-called "Man(cid:173) hattan world" assumption about the scene statistics of city and indoor scenes. This assumption stated that such scenes were built on a cartesian grid which led to regularities in the image edge gra(cid:173) dient statistics. In this paper we explore the general applicability of this assumption and show that, surprisingly, it holds in a large variety of less structured environments including rural scenes. This enables us, from a single image, to determine the orientation of the viewer relative to the scene structure and also to detect target ob(cid:173) jects which are not aligned with the grid. These inferences are performed using a Bayesian model with probability distributions (e.g. on the image gradient statistics) learnt from real data.

Variational approximations are becoming a widespread tool for Bayesian learning of graphical models. We provide some theoret(cid:173) ical results for the variational updates in a very general family of conjugate-exponential graphical models. We show how the belief propagation and the junction tree algorithms can be used in the inference step of variational Bayesian learning. Applying these re(cid:173) sults to the Bayesian analysis of linear-Gaussian state-space models we obtain a learning procedure that exploits the Kalman smooth(cid:173) ing propagation, while integrating over all model parameters. We demonstrate how this can be used to infer the hidden state dimen(cid:173) sionality of the state-space model in a variety of synthetic problems and one real high-dimensional data set.

The goal of many unsupervised learning procedures is to bring two probability distributions into alignment. Generative models such as Gaussian mixtures and Boltzmann machines can be cast in this light, as can recoding models such as ICA and projection pursuit. We propose a novel sample-based error measure for these classes of models, which applies even in situations where maximum likelihood (ML) and probability density estimation-based formulations can(cid:173) not be applied, e.g., models that are nonlinear or have intractable posteriors. Furthermore, our sample-based error measure avoids the difficulties of approximating a density function. We prove that with an unconstrained model, (1) our approach converges on the correct solution as the number of samples goes to infinity, and (2) the expected solution of our approach in the generative framework is the ML solution.

Jensen's inequality is a powerful mathematical tool and one of the workhorses in statistical learning. Its applications therein include the EM algorithm, Bayesian estimation and Bayesian inference. Quite often (i.e. in discriminative learning) upper bounds are needed as well. We derive and prove an efficient analytic inequality that provides such variational upper bounds. This inequality holds for latent variable mixtures of exponential family distributions and thus spans a wide range of contemporary statis(cid:173) tical models.