Bayesian Inference with Posterior Regularization and applications to Infinite Latent SVMs

Existing Bayesian models, especially nonparametric Bayesian methods, rely on specially conceived priors to incorporate domain knowledge for discovering improved latent representations. While priors can affect posterior distributions through Bayes' rule, imposing posterior regularization is arguably more direct and in some cases more natural and general. In this paper, we present regularized Bayesian inference (RegBayes), a novel computational framework that performs posterior inference with a regularization term on the desired post-data posterior distribution under an information theoretical formulation. RegBayes is more flexible than the procedure that elicits expert knowledge via priors, and it covers both directed Bayesian networks and undirected Markov networks whose Bayesian formulation results in hybrid chain graph models. When the regularization is induced from a linear operator on the posterior distributions, such as the expectation operator, we present a general convex-analysis theorem to characterize the solution of RegBayes. Furthermore, we present two concrete examples of RegBayes, infinite latent support vector machines (iLSVM) and multi-task infinite latent support vector machines (MT-iLSVM), which explore the large-margin idea in combination with a nonparametric Bayesian model for discovering predictive latent features for classification and multi-task learning, respectively. We present efficient inference methods and report empirical studies on several benchmark datasets, which appear to demonstrate the merits inherited from both large-margin learning and Bayesian nonparametrics. Such results were not available until now, and contribute to push forward the interface between these two important subfields, which have been largely treated as isolated in the community.Comment: 49 pages, 11 figure

Fraud/Uncollectible Debt Detection Using a Bayesian Network Based Learning System: A Rare Binary Outcome with Mixed Data Structures

The fraud/uncollectible debt problem in the telecommunications industry presents two technical challenges: the detection and the treatment of the account given the detection. In this paper, we focus on the first problem of detection using Bayesian network models, and we briefly discuss the application of a normative expert system for the treatment at the end. We apply Bayesian network models to the problem of fraud/uncollectible debt detection for telecommunication services. In addition to being quite successful at predicting rare event outcomes, it is able to handle a mixture of categorical and continuous data. We present a performance comparison using linear and non-linear discriminant analysis, classification and regression trees, and Bayesian network modelsComment: Appears in Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence (UAI1995

On data analysis and variable selection: the minimum entropy analysis

In this work, we present a minimum entropy analysis scheme for variable selection and preliminary data analysis. The variable selection can be achieved by the increasing preference of variables. We show such a preference to has a unqiue form, which is given by the entropy of models associated with variables. Evaluating the entropy provides a complete ranking scheme of variables. This scheme not only indicates preferred variables but also may reveal the system's nature and properties. We illustrate the proposed scheme to analyze a set of geological data for three carbonate rock units in Texas and Oklahoma, and compare to the discriminant function analysis. The result suggests this scheme to provide a quick and robust analysis, and the use in data analysis is promising.Comment: 9 pages and 2 table

Estimating Information-Theoretic Quantities with Uncertainty Forests

Information-theoretic quantities, such as conditional entropy and mutual information, are critical data summaries for quantifying uncertainty. Existing estimators for these quantities either have strong theoretical guarantees or effective performance in high-dimensional data, but not both. We propose a decision forest method, Uncertainty Forests (UF), which combines quantile regression forests, honest sampling, and a finite sample correction. We prove UF provides consistent estimates for these information-theoretic quantities, including in multivariate settings. Empirically, UF reduces finite sample bias and variance in a range of both low- and high-dimensional simulated settings for estimating posterior probabilities, conditional entropies, and mutual information. In a real-world connectome application, UF quantifies the uncertainty about neuron type given various cellular features in the Drosophila larva mushroom body, a key challenge for modern neuroscience

An eigenanalysis of data centering in machine learning

Many pattern recognition methods rely on statistical information from centered data, with the eigenanalysis of an empirical central moment, such as the covariance matrix in principal component analysis (PCA), as well as partial least squares regression, canonical-correlation analysis and Fisher discriminant analysis. Recently, many researchers advocate working on non-centered data. This is the case for instance with the singular value decomposition approach, with the (kernel) entropy component analysis, with the information-theoretic learning framework, and even with nonnegative matrix factorization. Moreover, one can also consider a non-centered PCA by using the second-order non-central moment. The main purpose of this paper is to bridge the gap between these two viewpoints in designing machine learning methods. To provide a study at the cornerstone of kernel-based machines, we conduct an eigenanalysis of the inner product matrices from centered and non-centered data. We derive several results connecting their eigenvalues and their eigenvectors. Furthermore, we explore the outer product matrices, by providing several results connecting the largest eigenvectors of the covariance matrix and its non-centered counterpart. These results lay the groundwork to several extensions beyond conventional centering, with the weighted mean shift, the rank-one update, and the multidimensional scaling. Experiments conducted on simulated and real data illustrate the relevance of this work.Comment: 14 page

The Importance of Being Clustered: Uncluttering the Trends of Statistics from 1970 to 2015

In this paper we retrace the recent history of statistics by analyzing all the papers published in five prestigious statistical journals since 1970, namely: Annals of Statistics, Biometrika, Journal of the American Statistical Association, Journal of the Royal Statistical Society, series B and Statistical Science. The aim is to construct a kind of "taxonomy" of the statistical papers by organizing and by clustering them in main themes. In this sense being identified in a cluster means being important enough to be uncluttered in the vast and interconnected world of the statistical research. Since the main statistical research topics naturally born, evolve or die during time, we will also develop a dynamic clustering strategy, where a group in a time period is allowed to migrate or to merge into different groups in the following one. Results show that statistics is a very dynamic and evolving science, stimulated by the rise of new research questions and types of data

Nonparametric discriminant HMM and application to facial expression recognition

Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2009, p. 2090-2096This paper presents a nonparametric discriminant HMM and applies it to facial expression recognition. In the proposed HMM, we introduce an effective nonparametric output probability estimation method to increase the discrimination ability at both hidden state level and class level. The proposed method uses a nonparametric adaptive kernel to utilize information from all classes and improve the discrimination at class level. The discrimination between hidden states is increased by defining membership coefficients which associate each reference vector with hidden states. The adaption of such coefficients is obtained by the Expectation Maximization (EM) method. Furthermore, we present a general formula for the estimation of output probability, which provides a way to develop new HMMs. Finally, we evaluate the performance of the proposed method on the CMU expression database and compare it with other nonparametric HMMs. © 2009 IEEE.published_or_final_versio

Stable Estimation of a Covariance Matrix Guided by Nuclear Norm Penalties

Estimation of covariance matrices or their inverses plays a central role in many statistical methods. For these methods to work reliably, estimated matrices must not only be invertible but also well-conditioned. In this paper we present an intuitive prior that shrinks the classic sample covariance estimator towards a stable target. We prove that our estimator is consistent and asymptotically efficient. Thus, it gracefully transitions towards the sample covariance matrix as the number of samples grows relative to the number of covariates. We also demonstrate the utility of our estimator in two standard situations -- discriminant analysis and EM clustering -- when the number of samples is dominated by or comparable to the number of covariates.Comment: 25 pages, 3 figure

Large-Scale Mode Identification and Data-Driven Sciences

Bump-hunting or mode identification is a fundamental problem that arises in almost every scientific field of data-driven discovery. Surprisingly, very few data modeling tools are available for automatic (not requiring manual case-by-base investigation), objective (not subjective), and nonparametric (not based on restrictive parametric model assumptions) mode discovery, which can scale to large data sets. This article introduces LPMode--an algorithm based on a new theory for detecting multimodality of a probability density. We apply LPMode to answer important research questions arising in various fields from environmental science, ecology, econometrics, analytical chemistry to astronomy and cancer genomics.Comment: I would like to express my sincere thanks to the Editor and the anonymous reviewers for their in-depth comments, which have greatly improved the manuscrip

IMMIGRATE: A Margin-based Feature Selection Method with Interaction Terms

Relief based algorithms have often been claimed to uncover feature interactions. However, it is still unclear whether and how interaction terms will be differentiated from marginal effects. In this paper, we propose IMMIGRATE algorithm by including and training weights for interaction terms. Besides applying the large margin principle, we focus on the robustness of the contributors of margin and consider local and global information simultaneously. Moreover, IMMIGRATE has been shown to enjoy attractive properties, such as robustness and combination with Boosting. We evaluate our proposed method on several tasks, which achieves state-of-the-art results significantly.Comment: R package ('Immigrate') available on CRA