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.

Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and deter(cid:173) mining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly represented by its own function approximator, indepen(cid:173) dent of the value function, and is updated according to the gradient of expected reward with respect to the policy parameters. Williams's REINFORCE method and actor-critic methods are examples of this approach. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. Using this result, we prove for the first time that a version of policy iteration with arbitrary differentiable function approximation is convergent to a locally optimal policy.

A new method for multivariate density estimation is developed based on the Support Vector Method (SVM) solution of inverse ill-posed problems. The solution has the form of a mixture of den(cid:173) sities. This method with Gaussian kernels compared favorably to both Parzen's method and the Gaussian Mixture Model method. For synthetic data we achieve more accurate estimates for densities of 2, 6, 12, and 40 dimensions.

We describe a joint probabilistic model for modeling the contents and inter-connectivity of document collections such as sets of web pages or research paper archives. The model is based on a probabilistic factor decomposition and allows identifying principal topics of the collection as well as authoritative documents within those topics. Furthermore, the relationships between topics is mapped out in order to build a predictive model of link content. Among the many applications of this approach are information retrieval and search, topic identification, query disambigua(cid:173) tion, focused web crawling, web authoring, and bibliometric analysis.

Learning of a smooth but nonparametric probability density can be reg(cid:173) ularized using methods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free pa(cid:173) rameter of the theory (,smoothness scale') can be determined self con(cid:173) sistently by the data; this forms an infinite dimensional generalization of the MDL principle. Finally, we study the implications of one's choice of the prior and the parameterization and conclude that the smoothness scale determination makes density estimation very weakly sensitive to the choice of the prior, and that even wrong choices can be advantageous for small data sets. One of the central problems in learning is to balance'goodness of fit' criteria against the complexity of models. An important development in the Bayesian approach was thus the realization that there does not need to be any extra penalty for model complexity: if we compute the total probability that data are generated by a model, there is a factor from the volume in parameter space-the'Occam factor' -that discriminates against models with more parameters [1, 2].

An important class of problems can be cast as inference in noisy(cid:173) OR Bayesian networks, where the binary state of each variable is a logical OR of noisy versions of the states of the variable's par(cid:173) ents. For example, in medical diagnosis, the presence of a symptom can be expressed as a noisy-OR of the diseases that may cause the symptom - on some occasions, a disease may fail to activate the symptom. Inference in richly-connected noisy-OR networks is in(cid:173) tractable, but approximate methods (e .g., variational techniques) are showing increasing promise as practical solutions. One prob(cid:173) lem with most approximations is that they tend to concentrate on a relatively small number of modes in the true posterior, ig(cid:173) noring other plausible configurations of the hidden variables. We introduce a new sequential variational method for bipartite noisy(cid:173) OR networks, that favors including all modes of the true posterior and models the posterior distribution as a tree.