Zoubin Ghahramani and Matthew J. Beal Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England {zoubin,m.beal}Ggatsby.ucl.ac.uk Abstract We present an algorithm that infers the model structure of a mixture offactor analysers using an efficient and deterministic variational approximationto full Bayesian integration over model parameters. Thisprocedure can automatically determine the optimal number of components and the local dimensionality of each component (Le. the number of factors in each factor analyser). Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to overfitting. Using a stochastic procedure for adding components it is possible to perform thevariational optimisation incrementally and to avoid local maxima.

Although one can derive the Gaussian noise assumption based on a maximum entropy approach, the main reason for this assumption is practicability: underthe Gaussian noise assumption the maximum likelihood parameter estimate can simply be found by minimization of the squared error. Despite its common use it is far from clear that the Gaussian noise assumption is a good choice for many practical problems. Areasonable approach therefore would be a noise distribution which contains the Gaussian as a special case but which has a tunable parameter that allows for more flexible distributions.

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.

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.

The first two methods are related to mean field ideas known in Statistical Physics. The third approach is based on Bayesian online approach which was motivated by recent results in the Statistical Mechanics of Neural Networks. We present simulation results showing: 1. that the mean field Bayesian evidence may be used for hyperparameter tuning and 2. that the online approach may achieve a low training error fast. 1 Introduction Gaussian processes provide promising nonparametric Bayesian approaches to regression andclassification [2, 1].

We study a population decoding paradigm in which the maximum likelihood inferenceis based on an unfaithful decoding model (UMLI). This is usually the case for neural population decoding because the encoding process of the brain is not exactly known, or because a simplified decoding modelis preferred for saving computational cost. We consider an unfaithful decoding model which neglects the pairwise correlation between neuronal activities, and prove that UMLI is asymptotically efficient whenthe neuronal correlation is uniform or of limited-range. The performance of UMLI is compared with that of the maximum likelihood inference based on a faithful model and that of the center of mass decoding method.It turns out that UMLI has advantages of decreasing the computational complexity remarkablely and maintaining a high-level decoding accuracy at the same time. The effect of correlation on the decoding accuracy is also discussed.

This article describes a methodology for programming robots known as probabilistic robotics. The probabilistic paradigm pays tribute to the inherent uncertainty in robot perception, relying on explicit representations of uncertainty when determining what to do. This article surveys some of the progress in the field, using in-depth examples to illustrate some of the nuts and bolts of the basic approach. My central conjecture is that the probabilistic approach to robotics scales better to complex real-world applications than approaches that ignore a robot's uncertainty.

Stochastic sampling algorithms, while an attractive alternative to exact algorithms in very large Bayesian network models, have been observed to perform poorly in evidential reasoning with extremely unlikely evidence. To address this problem, we propose an adaptive importance sampling algorithm, AIS-BN, that shows promising convergence rates even under extreme conditions and seems to outperform the existing sampling algorithms consistently. Three sources of this performance improvement are (1) two heuristics for initialization of the importance function that are based on the theoretical properties of importance sampling in finite-dimensional integrals and the structural advantages of Bayesian networks, (2) a smooth learning method for the importance function, and (3) a dynamic weighting function for combining samples from different stages of the algorithm. We tested the performance of the AIS-BN algorithm along with two state of the art general purpose sampling algorithms, likelihood weighting (Fung & Chang, 1989; Shachter & Peot, 1989) and self-importance sampling (Shachter & Peot, 1989). We used in our tests three large real Bayesian network models available to the scientific community: the CPCS network (Pradhan et al., 1994), the PathFinder network (Heckerman, Horvitz, & Nathwani, 1990), and the ANDES network (Conati, Gertner, VanLehn, & Druzdzel, 1997), with evidence as unlikely as 10^-41. While the AIS-BN algorithm always performed better than the other two algorithms, in the majority of the test cases it achieved orders of magnitude improvement in precision of the results. Improvement in speed given a desired precision is even more dramatic, although we are unable to report numerical results here, as the other algorithms almost never achieved the precision reached even by the first few iterations of the AIS-BN algorithm.

We show how to find a minimum weight loop cutset in a Bayesian network with high probability. Finding such a loop cutset is the first step in the method of conditioning for inference. Our randomized algorithm for finding a loop cutset outputs a minimum loop cutset after O(c 6^k kn) steps with probability at least 1 - (1 - 1/(6^k))^c6^k, where c > 1 is a constant specified by the user, k is the minimal size of a minimum weight loop cutset, and n is the number of vertices. We also show empirically that a variant of this algorithm often finds a loop cutset that is closer to the minimum weight loop cutset than the ones found by the best deterministic algorithms known.

Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameterless theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline for a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning, how the AIXI model can formally solve them. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXI-tl, which is still effectively more intelligent than any other time t and space l bounded agent. The computation time of AIXI-tl is of the order tx2^l. Other discussed topics are formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.