We propose a generative model for text and other collections of discrete data that generalizes or improves on several previous models including naive Bayes/unigram, mixture of unigrams [6], and Hof-mann's aspect model, also known as probabilistic latent semantic indexing (pLSI) [3]. In the context of text modeling, our model posits that each document is generated as a mixture of topics, where the continuous-valued mixture proportions are distributed as a latent Dirichlet random variable. Inference and learning are carried out efficiently via variational algorithms. We present empirical results on applications of this model to problems in text modeling, collaborative filtering, and text classification.

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Latent Dirichlet Allocation

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Table Of Integrals Series And Products

Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches.

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Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design

WINNER OF THE 2005 DEGROOT PRIZE! This book is for people who want to learn probability and statistics quickly. It brings together many of the main ideas in modern statistics in one place. The book is suitable for students and researchers in statistics, computer science, data mining and machine learning. This book covers a much wider range of topics than a typical introductory text on mathematical statistics. It includes modern topics like nonparametric curve estimation, bootstrapping and classification, topics that are usually relegated to follow-up courses. The reader is assumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. The text can be used at the advanced undergraduate and graduate level.

All of Statistics: A Concise Course in Statistical Inference

There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis obtainable to a wide audience. Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan provides an accessible approach to Bayesian data analysis, as material is explained clearly with concrete examples. The book begins with the basics, including essential concepts of probability and random sampling, and gradually progresses to advanced hierarchical modeling methods for realistic data. Included are step-by-step instructions on how to conduct Bayesian data analyses in the popular and free software R and WinBugs. This book is intended for first-year graduate students or advanced undergraduates. It provides a bridge between undergraduate training and modern Bayesian methods for data analysis, which is becoming the accepted research standard. Knowledge of algebra and basic calculus is a prerequisite. New to this Edition (partial list): * There are all new programs in JAGS and Stan. The new programs are designed to be much easier to use than the scripts in the first edition. In particular, there are now compact high-level scripts that make it easy to run the programs on your own data sets. This new programming was a major undertaking by itself.* The introductory Chapter 2, regarding the basic ideas of how Bayesian inference re-allocates credibility across possibilities, is completely rewritten and greatly expanded.* There are completely new chapters on the programming languages R (Ch. 3), JAGS (Ch. 8), and Stan (Ch. 14). The lengthy new chapter on R includes explanations of data files and structures such as lists and data frames, along with several utility functions. (It also has a new poem that I am particularly pleased with.) The new chapter on JAGS includes explanation of the RunJAGS package which executes JAGS on parallel computer cores. The new chapter on Stan provides a novel explanation of the concepts of Hamiltonian Monte Carlo. The chapter on Stan also explains conceptual differences in program flow between it and JAGS.* Chapter 5 on Bayes' rule is greatly revised, with a new emphasis on how Bayes' rule re-allocates credibility across parameter values from prior to posterior. The material on model comparison has been removed from all the early chapters and integrated into a compact presentation in Chapter 10.* What were two separate chapters on the Metropolis algorithm and Gibbs sampling have been consolidated into a single chapter on MCMC methods (as Chapter 7). There is extensive new material on MCMC convergence diagnostics in Chapters 7 and 8. There are explanations of autocorrelation and effective sample size. There is also exploration of the stability of the estimates of the HDI limits. New computer programs display the diagnostics, as well.* Chapter 9 on hierarchical models includes extensive new and unique material on the crucial concept of shrinkage, along with new examples.* All the material on model comparison, which was spread across various chapters in the first edition, in now consolidated into a single focused chapter (Ch. 10) that emphasizes its conceptualization as a case of hierarchical modeling.* Chapter 11 on null hypothesis significance testing is extensively revised. It has new material for introducing the concept of sampling distribution. It has new illustrations of sampling distributions for various stopping rules, and for multiple tests.* Chapter 12, regarding Bayesian approaches to null value assessment, has new material about the region of practical equivalence (ROPE), new examples of accepting the null value by Bayes factors, and new explanation of the Bayes factor in terms of the Savage-Dickey method.* Chapter 13, regarding statistical power and sample size, has an extensive new section on sequential testing, and making the research goal be precision of estimation instead of rejecting or accepting a particular value.* Chapter 15, which introduces the generalized linear model, is fully revised, with more complete tables showing combinations of predicted and predictor variable types.* Chapter 16, regarding estimation of means, now includes extensive discussion of comparing two groups, along with explicit estimates of effect size.* Chapter 17, regarding regression on a single metric predictor, now includes extensive examples of robust regression in JAGS and Stan. New examples of hierarchical regression, including quadratic trend, graphically illustrate shrinkage in estimates of individual slopes and curvatures. The use of weighted data is also illustrated.* Chapter 18, on multiple linear regression, includes a new section on Bayesian variable selection, in which various candidate predictors are probabilistically included in the regression model.* Chapter 19, on one-factor ANOVA-like analysis, has all new examples, including a completely worked out example analogous to analysis of covariance (ANCOVA), and a new example involving heterogeneous variances.* Chapter 20, on multi-factor ANOVA-like analysis, has all new examples, including a completely worked out example of a split-plot design that involves a combination of a within-subjects factor and a between-subjects factor.* Chapter 21, on logistic regression, is expanded to include examples of robust logistic regression, and examples with nominal predictors.* There is a completely new chapter (Ch. 22) on multinomial logistic regression. This chapter fills in a case of the generalized linear model (namely, a nominal predicted variable) that was missing from the first edition.* Chapter 23, regarding ordinal data, is greatly expanded. New examples illustrate single-group and two-group analyses, and demonstrate how interpretations differ from treating ordinal data as if they were metric.* There is a new section (25.4) that explains how to model censored data in JAGS.* Many exercises are new or revised. * Accessible, including the basics of essential concepts of probability and random sampling* Examples with R programming language and JAGS software* Comprehensive coverage of all scenarios addressed by non-Bayesian textbooks: t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis)* Coverage of experiment planning* R and JAGS computer programming code on website* Exercises have explicit purposes and guidelines for accomplishment* Provides step-by-step instructions on how to conduct Bayesian data analyses in the popular and free software R and WinBugs

Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan

We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, log-spline models, Dirichlet processes and interval censoring.

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Convergence rates of posterior distributions

Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics.

Fundamentals of Nonparametric Bayesian Inference

A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In the recent years, efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative.

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Posterior consistency of dirichlet mixtures in density estimation

We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or location-scale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub-Gaussian tails. We obtain bounds for Hellinger bracketing entropies for this class, and from these bounds, we deduce the convergence rates of (sieve) MLEs in Hellinger distance. The rate turns out to be $(log n)^\kappa /\sqrt{n}$, where $\kappa \ge 1$ is a constant that depends on the type of mixtures and the choice of the sieve. Next, we consider a Dirichlet mixture of normals as a prior on the unknown density. We estimate the prior probability of a certain Kullback-Leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the Hellinger entropy and the concentration rate of the prior. The posterior distribution is also seen to converge at the rate $(log n)^\kappa /\sqrt{n}$, where $\kappa$ now depends on the tail behavior of the base measure of the Dirichlet process.

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Rates of convergence for Bayes and maximum likelihood estimation for mixture of normal densities

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.

Subhashis Ghosal

Papers

Convergence rates of posterior distributions

Fundamentals of Nonparametric Bayesian Inference

Posterior consistency of dirichlet mixtures in density estimation

Rates of convergence for Bayes and maximum likelihood estimation for mixture of normal densities

Convergence rates of posterior distributions for noniid observations