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Author: Hajime Yamato Publisher: Springer Nature ISBN: 9811569754 Category : Mathematics Languages : en Pages : 80
Book Description
This book focuses on the properties associated with the Dirichlet process, describing its use a priori for nonparametric inference and the Bayes estimate to obtain limits for the estimable parameter. It presents the limits and the well-known U- and V-statistics as a convex combination of U-statistics, and by investigating this convex combination, it demonstrates these three statistics. Next, the book notes that the Dirichlet process gives the discrete distribution with probability one, even if the parameter of the process is continuous. Therefore, there are duplications among the sample from the distribution, which are discussed. Because sampling from the Dirichlet process is described sequentially, it can be described equivalently by the Chinese restaurant process. Using this process, the Donnelly–Tavaré–Griffiths formulas I and II are obtained, both of which give the Ewens’ sampling formula. The book then shows the convergence and approximation of the distribution for its number of distinct components. Lastly, it explains the interesting properties of the Griffiths–Engen–McCloskey distribution, which is related to the Dirichlet process and the Ewens’ sampling formula.
Author: Hajime Yamato Publisher: Springer Nature ISBN: 9811569754 Category : Mathematics Languages : en Pages : 80
Book Description
This book focuses on the properties associated with the Dirichlet process, describing its use a priori for nonparametric inference and the Bayes estimate to obtain limits for the estimable parameter. It presents the limits and the well-known U- and V-statistics as a convex combination of U-statistics, and by investigating this convex combination, it demonstrates these three statistics. Next, the book notes that the Dirichlet process gives the discrete distribution with probability one, even if the parameter of the process is continuous. Therefore, there are duplications among the sample from the distribution, which are discussed. Because sampling from the Dirichlet process is described sequentially, it can be described equivalently by the Chinese restaurant process. Using this process, the Donnelly–Tavaré–Griffiths formulas I and II are obtained, both of which give the Ewens’ sampling formula. The book then shows the convergence and approximation of the distribution for its number of distinct components. Lastly, it explains the interesting properties of the Griffiths–Engen–McCloskey distribution, which is related to the Dirichlet process and the Ewens’ sampling formula.
Author: Shuhei Mano Publisher: Springer ISBN: 4431558888 Category : Mathematics Languages : en Pages : 140
Book Description
This book focuses on statistical inferences related to various combinatorial stochastic processes. Specifically, it discusses the intersection of three subjects that are generally studied independently of each other: partitions, hypergeometric systems, and Dirichlet processes. The Gibbs partition is a family of measures on integer partition, and several prior processes, such as the Dirichlet process, naturally appear in connection with infinite exchangeable Gibbs partitions. Examples include the distribution on a contingency table with fixed marginal sums and the conditional distribution of Gibbs partition given the length. The A-hypergeometric distribution is a class of discrete exponential families and appears as the conditional distribution of a multinomial sample from log-affine models. The normalizing constant is the A-hypergeometric polynomial, which is a solution of a system of linear differential equations of multiple variables determined by a matrix A, called A-hypergeometric system. The book presents inference methods based on the algebraic nature of the A-hypergeometric system, and introduces the holonomic gradient methods, which numerically solve holonomic systems without combinatorial enumeration, to compute the normalizing constant. Furher, it discusses Markov chain Monte Carlo and direct samplers from A-hypergeometric distribution, as well as the maximum likelihood estimation of the A-hypergeometric distribution of two-row matrix using properties of polytopes and information geometry. The topics discussed are simple problems, but the interdisciplinary approach of this book appeals to a wide audience with an interest in statistical inference on combinatorial stochastic processes, including statisticians who are developing statistical theories and methodologies, mathematicians wanting to discover applications of their theoretical results, and researchers working in various fields of data sciences.
Author: Maya R. Gupta Publisher: Now Publishers Inc ISBN: 1601984308 Category : Computers Languages : en Pages : 87
Book Description
Introduces the expectation-maximization (EM) algorithm and provides an intuitive and mathematically rigorous understanding of this method. Theory and Use of the EM Algorithm is designed to be useful to both the EM novice and the experienced EM user looking to better understand the method and its use.
Author: Shui Feng Publisher: Springer ISBN: 9783642263798 Category : Mathematics Languages : en Pages : 0
Book Description
Presenting a comprehensive study of the Poisson-Dirichlet distribution, this volume emphasizes recent progress in evolutionary dynamics and asymptotic behaviors. The self-contained text presents methods and techniques that appeal to researchers in a wide variety of subjects.
Author: Nils Lid Hjort Publisher: Cambridge University Press ISBN: 1139484605 Category : Mathematics Languages : en Pages : 309
Book Description
Bayesian nonparametrics works - theoretically, computationally. The theory provides highly flexible models whose complexity grows appropriately with the amount of data. Computational issues, though challenging, are no longer intractable. All that is needed is an entry point: this intelligent book is the perfect guide to what can seem a forbidding landscape. Tutorial chapters by Ghosal, Lijoi and Prünster, Teh and Jordan, and Dunson advance from theory, to basic models and hierarchical modeling, to applications and implementation, particularly in computer science and biostatistics. These are complemented by companion chapters by the editors and Griffin and Quintana, providing additional models, examining computational issues, identifying future growth areas, and giving links to related topics. This coherent text gives ready access both to underlying principles and to state-of-the-art practice. Specific examples are drawn from information retrieval, NLP, machine vision, computational biology, biostatistics, and bioinformatics.
Author: Shui Feng Publisher: Springer Science & Business Media ISBN: 3642111947 Category : Mathematics Languages : en Pages : 228
Book Description
Presenting a comprehensive study of the Poisson-Dirichlet distribution, this volume emphasizes recent progress in evolutionary dynamics and asymptotic behaviors. The self-contained text presents methods and techniques that appeal to researchers in a wide variety of subjects.
Author: Jim Pitman Publisher: Springer Science & Business Media ISBN: 354030990X Category : Mathematics Languages : en Pages : 257
Book Description
The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes. There is particular focus on the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes.
Author: J.K. Ghosh Publisher: Springer Science & Business Media ISBN: 0387226540 Category : Mathematics Languages : en Pages : 311
Book Description
This book is the first systematic treatment of Bayesian nonparametric methods and the theory behind them. It will also appeal to statisticians in general. The book is primarily aimed at graduate students and can be used as the text for a graduate course in Bayesian non-parametrics.
Author: Dinh Phung Publisher: ISBN: Category : Correlation (Statistics) Languages : en Pages : 28
Book Description
"We introduce a new class of conditionally dependent Dirichlet processes (CDP) for hierarchical mixture modelling of naturally correlated data sources. This class of models provides a Bayesian nonparametric approach for modelling a range of challenging datasets which typically consists of heterogeneous observations from multiple correlated data channels. Some typical examples include annotated social media, networks in community where information about friendship and relation coexist with user's pro les, medical records where patient's information exists in several dimension (demographic information, medical history, drug uses and so on). The proposed framework can easily be tailored to model multiple data sources which are correlated by some latent underlying processes, whereas most of existing topic models, notably hierarchical Dirichlet processes (HDP), is designed for only a single data observation channel. In these existing approaches, data are grouped into documents (e.g., text documents or they are grouped according to some covariates such as time or location). Our approach is di erent: we view context as distributions over some index space and model both topics and contexts jointly. Distributions over topic parameters are modelled according to the usual Dirichlet processes. Stick-breaking representation gives rise to explicit realizations of topic atoms which we use as an indexing mechanism to induce conditional random mixture distributions on the context observation spaces { loosely speaking, we use a stochastic process, being DP, to conditionally `index' other stochastic processes. The later can be designed on any suitable family of stochastic processes to suit modelling needs or data types of contexts (such as Beta or Gaussian processes). Dirichlet process is of course an obvious choice. Our model can be viewed as an integration of the hierarchical Dirichlet process (HDP) and the recent nested Dirichlet process (nDP) with shared mixture components. In fact, it provides an interesting interpretation whereas, under a suitable parameterization, integrating out the topic components results in a nested DP, whereas integrating out the context components results in a hierarchical DP. Di erent approaches for posterior inference exist. This paper focus on the development of an auxiliary conditional Gibbs sampling in which both topic and context atoms are marginalized out. We demonstrate the framework on synthesis datasets for temporal topic modelling and trajectory discovery in videos surveillance. We then demonstrate an application on a current visual category classi cation challenge in computer vision for which we signi cantly outperform the current reported state-of-the-art results. Finally, it is worthwide to note that our proposed approach can be easily twisted to accommodate di erent forms of supervision (weakly annotated data, semi-supervision) and to perform prediction." -- Abstract.
Author: Hajime Yamato
Author: Hajime Yamato
Author: Shuhei Mano
Author: Maya R. Gupta
Author: Shui Feng
Author: Nils Lid Hjort
Author: Shui Feng
Author: Jim Pitman
Author: J.K. Ghosh
Author: Dev Basu
Author: Dinh Phung