In this paper we will discuss a variational mean-field theory and its application to BNs, sigmoidal BNs in particular. We present a variational derivation of the mean-field theory, proposed by Plefka[2].

The Bayesian paradigm apparently only sometimes gives rise to Occam's Razor; at other times very large models perform well. We give simple examples of both kinds of behaviour. The two views are reconciled when measuring complexity of functions, rather than of the machinery used to implement them. We analyze the complexity of functions for some linear in the parameter models that are equivalent to Gaussian Processes, and always find Occam's Razor at work. 1 Introduction Occam's Razor is a well known principle of "parsimony of explanations" which is influential inscientific thinking in general and in problems of statistical inference in particular. In this paper we review its consequences for Bayesian statistical models, where its behaviour can be easily demonstrated and quantified.

Learning of a smooth but nonparametric probability density can be regularized usingmethods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free parameter ofthe theory (,smoothness scale') can be determined self consistently bythe 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].

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. Jensen computes simplelower bounds on otherwise intractable quantities such as products of sums and latent log-likelihoods. This simplification then permits operationslike integration and maximization. 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 statistical models.We also discuss applications of the upper bounds including maximum conditional likelihood, large margin discriminative models and conditional Bayesian inference. Convergence, efficiency and prediction results are shown.

We use graphical models to explore the question of how people learn simple causalrelationships from data. The two leading psychological theories canboth be seen as estimating the parameters of a fixed graph. We argue that a complete account of causal induction should also consider how people learn the underlying causal graph structure, and we propose to model this inductive process as a Bayesian inference. Our argument is supported through the discussion of three data sets. 1 Introduction Causality plays a central role in human mental life. Our behavior depends upon our understanding ofthe causal structure of our environment, and we are remarkably good at inferring causation from mere observation. Constructing formal models of causal induction is currently a major focus of attention in computer science [7], psychology [3,6], and philosophy [5].This paper attempts to connect these literatures, by framing the debate between two major psychological theories in the computational language of graphical models. We show that existing theories equate human causal induction with maximum likelihood parameter estimationon a fixed graphical structure, and we argue that to fully account for human behavioral data, we must also postulate that people make Bayesian inferences about the underlying causal graph structure itself.

How should we decide among competing explanations of a cognitive process given limited observations? The problem of model selection is at the heart of progress in cognitive science. In this paper, Minimum Description Length (MDL) is introduced as a method for selecting among computational models of cognition. We also show that differential geometry provides an intuitive understanding of what drives model selection in MDL. Finally, adequacy of MDL is demonstrated in two areas of cognitive modeling.

The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields. The chances t_ij are usually estimated by the empirical sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual information. To answer questions like "is I(n_ij/n) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(t) comprising prior information about t. From the prior p(t) one can compute the posterior p(t|n), from which the distribution p(I|n) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(I|n). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.

I consider the problem of learning an optimal path graphical model from data and show the problem to be NP-hard for the maximum likelihood and minimum description length approaches and a Bayesian approach. This hardness result holds despite the fact that the problem is a restriction of the polynomially solvable problem of finding the optimal tree graphical model.

The chief aim of this paper is to propose mean-field approximations for a broad class of Belief networks, of which sigmoid and noisy-or networks can be seen as special cases. The approximations are based on a powerful mean-field theory suggested by Plefka. We show that Saul, Jaakkola and Jordan' s approach is the first order approximation in Plefka's approach, via a variational derivation. The application of Plefka's theory to belief networks is not computationally tractable. To tackle this problem we propose new approximations based on Taylor series. Small scale experiments show that the proposed schemes are attractive.

A general notion of algebraic conditional plausibility measures is defined. Probability measures, ranking functions, possibility measures, and (under the appropriate definitions) sets of probability measures can all be viewed as defining algebraic conditional plausibility measures. It is shown that algebraic conditional plausibility measures can be represented using Bayesian networks.