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.

We propose a new class of learning algorithms that combines variational approximation and Markov chain Monte Carlo (MCMC) simulation. Naive algorithms that use the variational approximation as proposal distribution can perform poorly because this approximation tends to underestimate the true variance and other features of the data. We solve this problem by introducing more sophisticated MCMC algorithms. One of these algorithms is a mixture of two MCMC kernels: a random walk Metropolis kernel and a blockMetropolis-Hastings (MH) kernel with a variational approximation as proposaldistribution. The MH kernel allows one to locate regions of high probability efficiently. The Metropolis kernel allows us to explore the vicinity of these regions. This algorithm outperforms variationalapproximations because it yields slightly better estimates of the mean and considerably better estimates of higher moments, such as covariances. It also outperforms standard MCMC algorithms because it locates theregions of high probability quickly, thus speeding up convergence. We demonstrate this algorithm on the problem of Bayesian parameter estimation for logistic (sigmoid) belief networks.

Determinantal point processes (DPPs) are elegant probabilistic models of repulsion that arise in quantum physics and random matrix theory. In contrast to traditional structured models like Markov random fields, which become intractable and hard to approximate in the presence of negative correlations, DPPs offer efficient and exact algorithms for sampling, marginalization, conditioning, and other inference tasks. We provide a gentle introduction to DPPs, focusing on the intuitions, algorithms, and extensions that are most relevant to the machine learning community, and show how DPPs can be applied to real-world applications like finding diverse sets of high-quality search results, building informative summaries by selecting diverse sentences from documents, modeling non-overlapping human poses in images or video, and automatically building timelines of important news stories.

We treat collaborative filtering as a univariate time series problem: given a user's previous votes, predict the next vote. We describe two families of methods for transforming data to encode time order in ways amenable to off-the-shelf classification and density estimation tools. Using a decision-tree learning tool and two real-world data sets, we compare the results of these approaches to the results of collaborative filtering without ordering information. The improvements in both predictive accuracy and in recommendation quality that we realize advocate the use of predictive algorithms exploiting the temporal order of data.

With the advance of efficient analytical methods for sensitivity analysis ofprobabilistic networks, the interest in the sensitivities revealed by real-life networks is rekindled. As the amount of data resulting from a sensitivity analysis of even a moderately-sized network is alreadyoverwhelming, methods for extracting relevant information are called for. One such methodis to study the derivative of the sensitivity functions yielded for a network's parameters. We further propose to build upon the concept of admissible deviation, that is, the extent to which a parameter can deviate from the true value without inducing a change in the most likely outcome. We illustrate these concepts by means of a sensitivity analysis of a real-life probabilistic network in oncology.

A Bayesian Belief Network (BN) is a model of a joint distribution over a setof n variables, with a DAG structure to represent the immediate dependenciesbetween the variables, and a set of parameters (aka CPTables) to represent thelocal conditional probabilities of a node, given each assignment to itsparents. In many situations, these parameters are themselves random variables - this may reflect the uncertainty of the domain expert, or may come from atraining sample used to estimate the parameter values. The distribution overthese "CPtable variables" induces a distribution over the response the BNwill return to any "What is Pr(H | E)?" query. This paper investigates thevariance of this response, showing first that it is asymptotically normal,then providing its mean and asymptotical variance. We then present aneffective general algorithm for computing this variance, which has the samecomplexity as simply computing the (mean value of) the response itself - ie,O(n 2^w), where n is the number of variables and w is the effective treewidth. Finally, we provide empirical evidence that this algorithm, whichincorporates assumptions and approximations, works effectively in practice,given only small samples.

We propose a new method of discovering causal structures, based on the detection of local, spontaneous changes in the underlying data-generating model. We analyze the classes of structures that are equivalent relative to a stream of distributions produced by local changes, and devise algorithms that output graphical representations of these equivalence classes. We present experimental results, using simulated data, and examine the errors associated with detection of changes and recovery of structures.

Chow and Liu (1968) studied the problem of learning a maximumlikelihood Markov tree. We generalize their work to more complexMarkov networks by considering the problem of learning a maximumlikelihood Markov network of bounded complexity. We discuss howtree-width is in many ways the appropriate measure of complexity andthus analyze the problem of learning a maximum likelihood Markovnetwork of bounded tree-width.Similar to the work of Chow and Liu, we are able to formalize thelearning problem as a combinatorial optimization problem on graphs. Weshow that learning a maximum likelihood Markov network of boundedtree-width is equivalent to finding a maximum weight hypertree. Thisequivalence gives rise to global, integer-programming based,approximation algorithms with provable performance guarantees, for thelearning problem. This contrasts with heuristic local-searchalgorithms which were previously suggested (e.g. by Malvestuto 1991).The equivalence also allows us to study the computational hardness ofthe learning problem. We show that learning a maximum likelihoodMarkov network of bounded tree-width is NP-hard, and discuss thehardness of approximation.