In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors to be a compactly parameterized product of Dirichlet distributions. Beside allowing for exact Bayesian learning, these results permit us to formulate a new class of tractable latent variable models in which the likelihood of a data point is computed through an ensemble average over tree structures.

Mixtures of Gaussians, factor analyzers (probabilistic PCA) and hidden Markov models are staples of static and dynamic data modeling and image and video modeling in particular. We show how topographic transformations in the input, such as translation and shearing in images, can be accounted for in these models by including a discrete transformation variable. The resulting models perform clustering, dimensionality reduction and time-series analysis in a way that is invariant to transformations in the input. Using the EM algorithm, these transformation-invariant models can be fit to static data and time series. We give results on filtering microscopy images, face and facial pose clustering, handwritten digit modeling and recognition, video clustering, object tracking, and removal of distractions from video sequences.

Recently developed techniques have made it possible to quickly learn accurate probability density functions from data in low-dimensional continuous space. In particular, mixtures of Gaussians can be fitted to data very quickly using an accelerated EM algorithm that employs multiresolution kd-trees (Moore, 1999). In this paper, we propose a kind of Bayesian networks in which low-dimensional mixtures of Gaussians over different subsets of the domain's variables are combined into a coherent joint probability model over the entire domain. The network is also capable of modeling complex dependencies between discrete variables and continuous variables without requiring discretization of the continuous variables. We present efficient heuristic algorithms for automatically learning these networks from data, and perform comparative experiments illustrated how well these networks model real scientific data and synthetic data. We also briefly discuss some possible improvements to the networks, as well as possible applications.

The K-Mean and EM algorithms are popular in clustering and mixture modeling, due to their simplicity and ease of implementation. However, they have several significant limitations. Both coverage to a local optimum of their respective objective functions (ignoring the uncertainty in the model space), require the apriori specification of the number of classes/clsuters, and are inconsistent. In this work we overcome these limitations by using the Minimum Message Length (MML) principle and a variation to the K-Means/EM observation assignment and parameter calculation scheme. We maintain the simplicity of these approaches while constructing a Bayesian mixture modeling tool that samples/searches the model space using a Markov Chain Monte Carlo (MCMC) sampler known as a Gibbs sampler. Gibbs sampling allows us to visit each model according to its posterior probability. Therefore, if the model space is multi-modal we will visit all models and not get stuck in local optima. We call our approach multiple chains at equilibrium (MCE) MML sampling.

Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chosen model will be used in the future for prediction tasks involving more ``focused' predictive distributions. Although focused predictive distributions can be produced from the joint probability distribution by marginalization, in practice the best model in the unsupervised sense does not necessarily perform well in supervised domains. In particular, the standard marginal likelihood score is a criterion for the unsupervised task, and, although frequently used for supervised model selection also, does not perform well in such tasks. In this paper we study the performance of the marginal likelihood score empirically in supervised Bayesian network selection tasks by using a large number of publicly available classification data sets, and compare the results to those obtained by alternative model selection criteria, including empirical crossvalidation methods, an approximation of a supervised marginal likelihood measure, and a supervised version of Dawids prequential(predictive sequential) principle.The results demonstrate that the marginal likelihood score does NOT perform well FOR supervised model selection, WHILE the best results are obtained BY using Dawids prequential r napproach.

The Support Vector Machine (SVM) of Vapnik (1998) has become widely established as one of the leading approaches to pattern recognition and machine learning. It expresses predictions in terms of a linear combination of kernel functions centred on a subset of the training data, known as support vectors. Despite its widespread success, the SVM suffers from some important limitations, one of the most significant being that it makes point predictions rather than generating predictive distributions. Recently Tipping (1999) has formulated the Relevance Vector Machine (RVM), a probabilistic model whose functional form is equivalent to the SVM. It achieves comparable recognition accuracy to the SVM, yet provides a full predictive distribution, and also requires substantially fewer kernel functions. The original treatment of the RVM relied on the use of type II maximum likelihood (the `evidence framework') to provide point estimates of the hyperparameters which govern model sparsity. In this paper we show how the RVM can be formulated and solved within a completely Bayesian paradigm through the use of variational inference, thereby giving a posterior distribution over both parameters and hyperparameters. We demonstrate the practicality and performance of the variational RVM using both synthetic and real world examples.

We propose a novel reversible jump Markov chain Monte Carlo (MCMC) simulated annealing algorithm to optimize radial basis function (RBF) networks. This algorithm enables us to maximize the joint posterior distribution of the network parameters and the number of basis functions. It performs a global search in the joint space of the parameters and number of parameters, thereby surmounting the problem of local minima. We also show that by calibrating a Bayesian model, we can obtain the classical AIC, BIC and MDL model selection criteria within a penalized likelihood framework. Finally, we show theoretically and empirically that the algorithm converges to the modes of the full posterior distribution in an efficient way.

Finite mixture modeling is a flexible and powerful approach to modeling data that is heterogeneous and stems from multiple populations, such as data from patter recognition, computer vision, image analysis, and machine learning. The Gaussian mixture model is an important mixture model family. It is well known that any continuous distribution can be approximated arbitrarily well by a finite mixture of normal densities (Lindsay, 1995; McLachlan and Peel, 2000). However, as demonstrated by Chen (1995), when the number of components is unknown, the optimal convergence rate of the estimate of a finite mixture model is slower than the optimal convergence rate when the number is known. In practice, with too many components, the mixture may overfit the data and yield poor interpretations, while with too few components, the mixture may not be flexible enough to approximate the true underlying data structure.

We present an iterative Markov chain Monte Carlo algorithm for computing reference priors and minimax risk for general parametric families. Our approach uses MCMC techniques based on the Blahut-Arimoto algorithm for computing channel capacity in information theory. We give a statistical analysis of the algorithm, bounding the number of samples required for the stochastic algorithm to closely approximate the deterministic algorithm in each iteration. Simulations are presented for several examples from exponential families. Although we focus on applications to reference priors and minimax risk, the methods and analysis we develop are applicable to a much broader class of optimization problems and iterative algorithms.

It has been argued that in supervised classification tasks, in practice it may be more sensible to perform model selection with respect to some more focused model selection score, like the supervised (conditional) marginal likelihood, than with respect to the standard marginal likelihood criterion. However, for most Bayesian network models, computing the supervised marginal likelihood score takes exponential time with respect to the amount of observed data. In this paper, we consider diagnostic Bayesian network classifiers where the significant model parameters represent conditional distributions for the class variable, given the values of the predictor variables, in which case the supervised marginal likelihood can be computed in linear time with respect to the data. As the number of model parameters grows in this case exponentially with respect to the number of predictors, we focus on simple diagnostic models where the number of relevant predictors is small, and suggest two approaches for applying this type of models in classification. The first approach is based on mixtures of simple diagnostic models, while in the second approach we apply the small predictor sets of the simple diagnostic models for augmenting the Naive Bayes classifier.