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Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

444 records · Page 25Linked to original sources
arXiv2011

Leo Breiman

Statistics is a uniquely difficult field to convey to the uninitiated. It sits astride the abstract and the concrete, the theoretical and the applied. It has a mathematical flavor and yet it is not simply a branch of mathematics. Its core problems blend into those of the disciplines that probe into the nature of intelligence and thought, in particular philosophy, psychology and artificial intelligence. Debates over foundational issues have waxed and waned, but the field has not yet arrived at a single foundational perspective.

stat.AP↗
arXiv2010

Heavy-Tailed Processes for Selective Shrinkage

Heavy-tailed distributions are frequently used to enhance the robustness of regression and classification methods to outliers in output space. Often, however, we are confronted with "outliers" in input space, which are isolated observations in sparsely populated regions. We show that heavy-tailed stochastic processes (which we construct from Gaussian processes via a copula), can be used to improve robustness of regression and classification estimators to such outliers by selectively shrinking them more strongly in sparse regions than in dense regions. We carry out a theoretical analysis to show that selective shrinkage occurs, provided the marginals of the heavy-tailed process have sufficiently heavy tails. The analysis is complemented by experiments on biological data which indicate significant improvements of estimates in sparse regions while producing competitive results in dense regions.

stat.ML↗
arXiv2010

Tree-Structured Stick Breaking Processes for Hierarchical Data

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

stat.ME↗
arXiv2010

Bayesian Nonparametric Inference of Switching Linear Dynamical Systems

Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index, and a maneuvering target tracking application.

stat.ME↗
arXiv2009

The nested Chinese restaurant process and Bayesian nonparametric inference of topic hierarchies

We present the nested Chinese restaurant process (nCRP), a stochastic process which assigns probability distributions to infinitely-deep, infinitely-branching trees. We show how this stochastic process can be used as a prior distribution in a Bayesian nonparametric model of document collections. Specifically, we present an application to information retrieval in which documents are modeled as paths down a random tree, and the preferential attachment dynamics of the nCRP leads to clustering of documents according to sharing of topics at multiple levels of abstraction. Given a corpus of documents, a posterior inference algorithm finds an approximation to a posterior distribution over trees, topics and allocations of words to levels of the tree. We demonstrate this algorithm on collections of scientific abstracts from several journals. This model exemplifies a recent trend in statistical machine learning--the use of Bayesian nonparametric methods to infer distributions on flexible data structures.

stat.ML↗
arXiv2009

Kernel dimension reduction in regression

We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace [cf. J. Amer. Statist. Assoc. 86 (1991) 316--342 and Regression Graphics (1998) Wiley]. We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an $M$-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.

math.ST↗
arXiv2009

Estimating divergence functionals and the likelihood ratio by convex risk minimization

We develop and analyze $M$-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which allows the problem of estimating divergences to be tackled via convex empirical risk optimization. The resulting estimators are simple to implement, requiring only the solution of standard convex programs. We present an analysis of consistency and convergence for these estimators. Given conditions only on the ratios of densities, we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes. We derive an efficient optimization algorithm for computing our estimates, and illustrate their convergence behavior and practical viability by simulations.

math.ST↗
arXiv2009

On surrogate loss functions and $f$-divergences

The goal of binary classification is to estimate a discriminant function $γ$ from observations of covariate vectors and corresponding binary labels. We consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality-reducing quantizer $Q$. We present conditions on loss functions such that empirical risk minimization yields Bayes consistency when both the discriminant function and the quantizer are estimated. These conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as Ali-Silvey or $f$-divergence functionals. Whereas this correspondence was established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951) 93--102. Univ. California Press, Berkeley] for the 0--1 loss, we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of Bayes consistency for binary classification. Our result makes it possible to pick out the (strict) subset of surrogate loss functions that yield Bayes consistency for joint estimation of the discriminant function and the quantizer.

math.ST↗
arXiv2008

On optimal quantization rules for some problems in sequential decentralized detection

We consider the design of systems for sequential decentralized detection, a problem that entails several interdependent choices: the choice of a stopping rule (specifying the sample size), a global decision function (a choice between two competing hypotheses), and a set of quantization rules (the local decisions on the basis of which the global decision is made). This paper addresses an open problem of whether in the Bayesian formulation of sequential decentralized detection, optimal local decision functions can be found within the class of stationary rules. We develop an asymptotic approximation to the optimal cost of stationary quantization rules and exploit this approximation to show that stationary quantizers are not optimal in a broad class of settings. We also consider the class of blockwise stationary quantizers, and show that asymptotically optimal quantizers are likelihood-based threshold rules.

math.ST↗
arXiv2006

A direct formulation for sparse PCA using semidefinite programming

We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. The problem arises in the decomposition of a covariance matrix into sparse factors, and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the SDP arising in the direct sparse PCA method.

cs.CE↗
arXiv2004

Subtree power analysis finds optimal species for comparative genomics

Sequence comparison across multiple organisms aids in the detection of regions under selection. However, resource limitations require a prioritization of genomes to be sequenced. This prioritization should be grounded in two considerations: the lineal scope encompassing the biological phenomena of interest, and the optimal species within that scope for detecting functional elements. We introduce a statistical framework for optimal species subset selection, based on maximizing power to detect conserved sites. In a study of vertebrate species, we show that the optimal species subset is not in general the most evolutionarily diverged subset. Our results suggest that marsupials are prime sequencing candidates.

q-bio.GN↗