We propose a novel "tree-averaging" model that utilizes the ensemble of classification and regression trees (CART). Each constituent tree is estimated with a subset of similar data. We treat this grouping of subsets as Bayesian ensemble trees (BET) and model them as an infinite mixture Dirichlet process. We show that BET adapts to data heterogeneity and accurately estimates each component. Compared with the bootstrap-aggregating approach, BET shows improved prediction performance with fewer trees. We develop an efficient estimating procedure with improved sampling strategies in both CART and mixture models. We demonstrate these advantages of BET with simulations, classification of breast cancer and regression of lung function measurement of cystic fibrosis patients.

We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the other to gradient matching. All methods have greater accuracy than previously suggested Gaussian Process approaches. We also suggest a general approach that can yield error estimates from any standard ODE solver.

While labeled data is expensive to prepare, ever increasing amounts of unlabeled data is becoming widely available. In order to adapt to this phenomenon, several semi-supervised learning (SSL) algorithms, which learn from labeled as well as unlabeled data, have been developed. In a separate line of work, researchers have started to realize that graphs provide a natural way to represent data in a variety of domains. Graph-based SSL algorithms, which bring together these two lines of work, have been shown to outperform the state-of-the-art in many applications in speech processing, computer vision, natural language processing, and other areas of Artificial Intelligence. Recognizing this promising and emerging area of research, this synthesis lecture focuses on graph-based SSL algorithms (e.g., label propagation methods).

Deeply rooted in classical social choice and voting theory, statistical ranking with paired comparison data experienced its renaissance with the wide spread of crowdsourcing technique. As the data quality might be significantly damaged in an uncontrolled crowdsourcing environment, outlier detection and robust ranking have become a hot topic in such data analysis. In this paper, we propose a robust ranking framework based on the principle of Huber's robust statistics, which formulates outlier detection as a LASSO problem to find sparse approximations of the cyclic ranking projection in Hodge decomposition. Moreover, simple yet scalable algorithms are developed based on Linearized Bregman Iteration to achieve an even less biased estimator than LASSO. Statistical consistency of outlier detection is established in both cases which states that when the outliers are strong enough and in Erdos-Renyi random graph sampling settings, outliers can be faithfully detected. Our studies are supported by experiments with both simulated examples and real-world data. The proposed framework provides us a promising tool for robust ranking with large scale crowdsourcing data arising from computer vision, multimedia, machine learning, sociology, etc.

Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply l1-penalties to either (1) parametric likelihoods, or, (2) regularized regression/pseudo-likelihoods, with the latter having the distinct advantage that they do not explicitly assume Gaussianity. As none of the popular methods proposed for solving pseudo-likelihood based objective functions have provable convergence guarantees, it is not clear if corresponding estimators exist or are even computable, or if they actually yield correct partial correlation graphs. This paper proposes a new pseudo-likelihood based graphical model selection method that aims to overcome some of the shortcomings of current methods, but at the same time retain all their respective strengths. In particular, we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem, resulting in an objective function comprised of quadratic forms. The objective is then optimized via a coordinate-wise approach. The specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees; an important property that cannot be established using standard results, when the dimension is larger than the sample size, as is often the case in high dimensional applications. These convergence guarantees ensure that estimators are well-defined under very general conditions, and are always computable. In addition, the approach yields estimators that have good large sample properties and also respect symmetry. Furthermore, application to simulated/real data, timing comparisons and numerical convergence is demonstrated. We also present a novel unifying framework that places all graphical pseudo-likelihood methods as special cases of a more general formulation, leading to important insights.

MachineLearning manuscript No. (will be inserted by the editor) Abstract We discuss the problem of risk estimation in the classification problem, with specific focus on finding distributions that maximize the confidence intervals of risk estimation. We derived simple analytic approximations for the maximum bias of empirical risk for histogram classifier. We carry out a detailed study on using these analytic estimates for empirical estimation of risk. Keywords data mining · machine learning · misclassification probability · overfitting · confidence interval · statistical estimate 1 Introduction The study of overfitting is one of the most important research directions in the area of machine learning. This problem arises from common disadvantage of more complex decision rules relative to the simpler ones when the sample size is not very large.

Where Pa, are the parent variables of the vertex i and dY is the data restricted to the coordinates in Y Q X. A Bayesian approach to structure discovery in Bayesian networks.

We introduce statistical constraints, a declarative modelling tool that links statistics and constraint programming. We discuss two statistical constraints and some associated filtering algorithms. Finally, we illustrate applications to standard problems encountered in statistics and to a novel inspection scheduling problem in which the aim is to find inspection plans with desirable statistical properties.

In this paper, we investigate the statistical convergence rate of a Bayesian low-rank tensor estimator. Our problem setting is the regression problem where a tensor structure underlying the data is estimated. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a near optimal rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori.

Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that storing and computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices, when combined with diagonal Gaussian matrices, exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks proposed by Rahimi and Recht (2009) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy. Our method applies to any translation invariant and any dot-product kernel, such as the popular RBF kernels and polynomial kernels. We prove that the approximation is unbiased and has low variance. Experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and/or require real-time prediction.