In a Bayesian mixture model it is not necessary a priori to limit the num(cid:173) ber of components to be finite. In this paper an infinite Gaussian mixture model is presented which neatly sidesteps the difficult problem of find(cid:173) ing the "right" number of mixture components. Inference in the model is done using an efficient parameter-free Markov Chain that relies entirely on Gibbs sampling.

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.

By thinking of each state in a hidden Markov model as corresponding to some spatial region of a fictitious topology space it is possible to naturally define neigh(cid:173) bouring states as those which are connected in that space. The transition matrix can then be constrained to allow transitions only between neighbours; this means that all valid state sequences correspond to connected paths in the topology space. I show how such constrained HMMs can learn to discover underlying structure in complex sequences of high dimensional data, and apply them to the problem of recovering mouth movements from acoustics in continuous speech. Probabilistic unsupervised learning for such sequences requires models with two essential features: latent (hidden) variables and topology in those variables. Hidden Markov models (HMMs) can be thought of as dynamic generalizations of discrete state static data models such as Gaussian mixtures, or as discrete state versions of linear dynam(cid:173) ical systems (LDSs) (which are themselves dynamic generalizations of continuous latent variable models such as factor analysis).

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.

We first show how to represent sharp posterior probability distribu(cid:173) tions using real valued coefficients on broadly-tuned basis functions. Then we show how the precise times of spikes can be used to con(cid:173) vey the real-valued coefficients on the basis functions quickly and accurately. Finally we describe a simple simulation in which spik(cid:173) ing neurons learn to model an image sequence by fitting a dynamic generative model. A perceived object is represented in the brain by the activities of many neurons, but there is no general consensus on how the activities of individual neurons combine to represent the multiple properties of an object. We start by focussing on the case of a single object that has multiple instantiation parameters such as position, velocity, size and orientation. We assume that each neuron has an ideal stimulus in the space of instantiation parameters and that its activation rate or probability of activation falls off monotonically in all directions as the actual stimulus departs from this ideal.

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.

We propose a new approach to the problem of searching a space of stochastic controllers for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP). Following several other authors, our approach is based on searching in parameterized families of policies (for example, via gradient descent) to optimize solution qual(cid:173) ity. However, rather than trying to estimate the values and derivatives of a policy directly, we do so indirectly using estimates for the proba(cid:173) bility densities that the policy induces on states at the different points in time. This enables our algorithms to exploit the many techniques for efficient and robust approximate density propagation in stochastic sys(cid:173) tems. We show how our techniques can be applied both to deterministic propagation schemes (where the MDP's dynamics are given explicitly in compact form,) and to stochastic propagation schemes (where we have access only to a generative model, or simulator, of the MDP).

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 present a Hidden Markov Model (HMM) for inferring the hidden psychological state (or neural activity) during single trial tMRI activa(cid:173) tion experiments with blocked task paradigms. Inference is based on Bayesian methodology, using a combination of analytical and a variety of Markov Chain Monte Carlo (MCMC) sampling techniques. The ad(cid:173) vantage of this method is that detection of short time learning effects be(cid:173) tween repeated trials is possible since inference is based only on single trial experiments.