The problem of reinforcement learning in a non-Markov environment is explored using a dynamic Bayesian network, where conditional independence assumptions between random variables are compactly represented by network parameters. The parameters are learned online, and approximations are used to perform inference and to compute the optimal value function. The relative effects of inference and value function approximations on the quality of the final policy are investigated, by learning to solve a moderately difficult driving task. The two value function approximations, linear and quadratic, were found to perform similarly, but the quadratic model was more sensitive to initialization. Both performed below the level of human performance on the task. The dynamic Bayesian network performed comparably to a model using a localist hidden state representation, while requiring exponentially fewer parameters.

Gaussian mixtures (or so-called radial basis function networks) for density estimation provide a natural counterpart to sigmoidal neural networks for function fitting and approximation. In both cases, it is possible to give simple expressions for the iterative improvement of performance as components of the network are introduced one at a time. In particular, for mixture density estimation we show that a k-component mixture estimated by maximum likelihood (or by an iterative likelihood improvement that we introduce) achieves log-likelihood within order 1/k of the log-likelihood achievable by any convex combination. Consequences for approximation and estimation using Kullback-Leibler risk are also given. A Minimum Description Length principle selects the optimal number of components k that minimizes the risk bound. 1 Introduction In density estimation, Gaussian mixtures provide flexible-basis representations for densities that can be used to model heterogeneous data in high dimensions. Consider a parametric family G { pe(x), x E X C Rd': fJ E The main theme of the paper is to give approximation and estimation bounds of arbitrary densities by finite mixture densities.

We propose and analyze a class of actor-critic algorithms for simulation-based optimization of a Markov decision process over a parameterized family of randomized stationary policies. These are two-time-scale algorithms in which the critic uses TD learning with a linear approximation architecture and the actor is updated in an approximate gradient direction based on information provided bythe critic. We show that the features for the critic should span a subspace prescribed by the choice of parameterization of the actor. We conclude by discussing convergence properties and some open problems.

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 neighbouring statesas 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.

We present a new technique for time series analysis based on dynamic probabilisticnetworks. In this approach, the observed data are modeled in terms of unobserved, mutually independent factors, as in the recently introduced technique of Independent Factor Analysis (IFA).However, unlike in IFA, the factors are not Li.d.; each factor has its own temporal statistical characteristics. We derive a family of EM algorithms that learn the structure of the underlying factors and their relation to the data. These algorithms perform source separation and noise reduction in an integrated manner, and demonstrate superior performance compared to IFA. 1 Introduction The technique of independent factor analysis (IFA) introduced in [1] provides a tool for modeling L'-dim data in terms of L unobserved factors. These factors are mutually independent and combine linearly with added noise to produce the observed data.

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 quality. However, rather than trying to estimate the values and derivatives of a policy directly, we do so indirectly using estimates for the probability 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 systems. 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).

We propose a novel approach for building finite memory predictive models similar in spirit to variable memory length Markov models (VLMMs). The models are constructed by first transforming the n-block structure of the training sequence into a spatial structure of points in a unit hypercube, such that the longer is the common suffix shared by any two n-blocks, the closer lie their point representations. Such a transformation embodies a Markov assumption - n-blocks with long common suffixes are likely to produce similar continuations. Finding a set of prediction contexts is formulated as a resource allocation problem solved by vector quantizing the spatial n-block representation. We compare our model with both the classical and variable memory length Markov models on three data sets with different memory and stochastic components. Our models have a superior performance, yet, their construction is fully automatic, which is shown to be problematic in the case of VLMMs.

We describe a Bayesian approach to model selection in unsupervised learning that determines both the feature set and the number of clusters. We then evaluate this scheme (based on marginal likelihood) and one based on cross-validated likelihood. For the Bayesian scheme we derive a closed-form solution of the marginal likelihood by assuming appropriate forms of the likelihood function and prior. Extensive experiments compare these approaches and all results are verified by comparison against ground truth. In these experiments the Bayesian scheme using our objective function gave better results than cross-validation. 1 Introduction Recent efforts define the model selection problem as one of estimating the number of clusters[ 10, 17].

In particular, Mackay showed that by approximating the distributions of the weights with Gaussians and adopting smoothing priors, it is possible to obtain estimates of the weights and output variances and to automatically set the regularisation coefficients. Neal (1996) cast the net much further by introducing advanced Bayesian simulation methods, specifically the hybrid Monte Carlo method, into the analysis of neural networks [3]. Bayesian sequential Monte Carlo methods have also been shown to provide good training results, especially in time-varying scenarios [4]. More recently, Rios Insua and Muller (1998) and Holmes and Mallick (1998) have addressed the issue of selecting the number of hidden neurons with growing and pruning algorithms from a Bayesian perspective [5,6]. In particular, they apply the reversible jump Markov Chain Monte Carlo (MCMC) algorithm of Green [7] to feed-forward sigmoidal networks and radial basis function (RBF) networks to obtain joint estimates of the number of neurons and weights. We also apply the reversible jump MCMC simulation algorithm to RBF networks so as to compute the joint posterior distribution of the radial basis parameters and the number of basis functions. However, we advance this area of research in two important directions. Firstly, we propose a full hierarchical prior for RBF networks.

We consider the problem of reconstructing a temporal discrete sequence of multidimensional real vectors when part of the data is missing, under the assumption that the sequence was generated by a continuous process. A particular case of this problem is multivariate regression, which is very difficult when the underlying mapping is one-to-many. We propose an algorithm based on a joint probability model of the variables of interest, implemented using a nonlinear latent variable model. Each point in the sequence is potentially reconstructed as any of the modes of the conditional distribution of the missing variables given the present variables (computed using an exhaustive mode search in a Gaussian mixture). Mode selection is determined by a dynamic programming search that minimises a geometric measure of the reconstructed sequence, derived from continuity constraints. We illustrate the algorithm with a toy example and apply it to a real-world inverse problem, the acoustic-toarticulatory mapping. The results show that the algorithm outperforms conditional mean imputation and multilayer perceptrons. 1 Definition of the problem