In many domains, we are interested in analyzing the structure of the underlying distribution, e.g., whether one variable is a direct parent of the other. Bayesian model-selection attempts to find the MAP model and use its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible posterior. Thus, we want compute the Bayesian posterior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to efficiently compute a sum over the exponential number of networks that are consistent with a fixed ordering over network variables. This allows us to compute, for a given ordering, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approximates the Bayesian posterior of a feature. Our approach uses a Markov Chain Monte Carlo (MCMC) method, but over orderings rather than over network structures. The space of orderings is much smaller and more regular than the space of structures, and has a smoother posterior `landscape'. We present empirical results on synthetic and real-life datasets that compare our approach to full model averaging (when possible), to MCMC over network structures, and to a non-Bayesian bootstrap approach.

This paper is about metric data structures in high-dimensional or non-Euclidean space that permit cached sufficient statistics accelerations of learning algorithms. It has recently been shown that for less than about 10 dimensions, decorating kd-trees with additional "cached sufficient statistics" such as first and second moments and contingency tables can provide satisfying acceleration for a very wide range of statistical learning tasks such as kernel regression, locally weighted regression, k-means clustering, mixture modeling and Bayes Net learning. In this paper, we begin by defining the anchors hierarchy - a fast data structure and algorithm for localizing data based only on a triangle-inequality-obeying distance metric. We show how this, in its own right, gives a fast and effective clustering of data. But more importantly we show how it can produce a well-balanced structure similar to a Ball-Tree (Omohundro, 1991) or a kind of metric tree (Uhlmann, 1991; Ciaccia, Patella, & Zezula, 1997) in a way that is neither "top-down" nor "bottom-up" but instead "middle-out". We then show how this structure, decorated with cached sufficient statistics, allows a wide variety of statistical learning algorithms to be accelerated even in thousands of dimensions.

We present an approach to model-based hierarchical clustering by formulating an objective function based on a Bayesian analysis. This model organizes the data into a cluster hierarchy while specifying a complex feature-set partitioning that is a key component of our model. Features can have either a unique distribution in every cluster or a common distribution over some (or even all) of the clusters. The cluster subsets over which these features have such a common distribution correspond to the nodes (clusters) of the tree representing the hierarchy. We apply this general model to the problem of document clustering for which we use a multinomial likelihood function and Dirichlet priors. Our algorithm consists of a two-stage process wherein we first perform a flat clustering followed by a modified hierarchical agglomerative merging process that includes determining the features that will have common distributions over the merged clusters. The regularization induced by using the marginal likelihood automatically determines the optimal model structure including number of clusters, the depth of the tree and the subset of features to be modeled as having a common distribution at each node. We present experimental results on both synthetic data and a real document collection.

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.

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.

The objective of change-point detection is to discover abrupt property changes lying behind time-series data. In this paper, we present a novel statistical change-point detection algorithm based on non-parametric divergence estimation between time-series samples from two retrospective segments. Our method uses the relative Pearson divergence as a divergence measure, and it is accurately and efficiently estimated by a method of direct density-ratio estimation. Through experiments on artificial and real-world datasets including human-activity sensing, speech, and Twitter messages, we demonstrate the usefulness of the proposed method.

Recent theoretical work has identified random projection as a promising dimensionality reduction technique for learning mixtures of Gausians. Here we summarize these results and illustrate them by a wide variety of experiments on synthetic and real data.

Incorporating feature selection into a classification or regression method often carries a number of advantages. In this paper we formalize feature selection specifically from a discriminative perspective of improving classification/regression accuracy. The feature selection method is developed as an extension to the recently proposed maximum entropy discrimination (MED) framework. We describe MED as a flexible (Bayesian) regularization approach that subsumes, e.g., support vector classification, regression and exponential family models. For brevity, we restrict ourselves primarily to feature selection in the context of linear classification/regression methods and demonstrate that the proposed approach indeed carries substantial improvements in practice. Moreover, we discuss and develop various extensions of feature selection, including the problem of dealing with example specific but unobserved degrees of freedom -- alignments or invariants.

We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain classifiers that optimize the cross-entropy function. In the possibilistic framework, we obtain classifiers that maximize the interclass margin. Furthermore, we show that the support vector machine is a sub-class of these maximum-margin classifiers.

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.