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Group representations in probability and statistics

01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1522 citations till now. The article focuses on the topics: Probability and statistics & Convolution of probability distributions.
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Book
01 Dec 2008
TL;DR: Markov Chains and Mixing Times as mentioned in this paper is an introduction to the modern approach to the theory of Markov chains and its application in the field of probability theory and linear algebra, where the main goal is to determine the rate of convergence of a Markov chain to the stationary distribution.
Abstract: This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

2,573 citations


Cites background from "Group representations in probabilit..."

  • ...The survey by Diaconis and SaloffCoste (1998) contains more on the Metropolis algorithm....

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  • ...The effectiveness of the `(2) bound was first demonstrated by Diaconis and Shahshahani (1981). Diaconis (1988) uses representation theory to calculate eigenvalues and eigenfunctions for random walks on groups. Spielman and Teng (1996) show that for any planar graph with n vertices and maximum degree ∆, the relaxation time is at least c(∆)n, where c(∆) is a constant depending on the ∆....

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  • ...The survey by Diaconis and SaloffCoste (1998) contains more on the Metropolis algorithm. The textbook Brémaud (1999) also discusses the use of the Metropolis algorithm for Monte Carlo sampling....

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  • ...In fact, as Diaconis and Shahshahani (1981) proved, the random transpositions walk has a sharp cutoff of width O(n) at (1/2)n log n....

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  • ...Diaconis (1988) is a starting place. Pólya’s urn was introduced in Eggenberger and Pólya (1923) and Pólya (1931). Urns are fundamental models for reinforced process. See Pemantle (2007) for a wealth of information and many references on urn processes and more generally processes with reinforcement....

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01 Jan 2001
TL;DR: Estimates on the important parameters of access time, commute time, cover time and mixing time are discussed and recent algorithmic applications of random walks are sketched, in particular to the problem of sampling.
Abstract: Various aspects of the theory of random walks on graphs are surveyed In particular, estimates on the important parameters of access time, commute time, cover time and mixing time are discussed Connections with the eigenvalues of graphs and with electrical networks, and the use of these connections in the study of random walks is described We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling

1,564 citations


Cites background from "Group representations in probabilit..."

  • ...We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling....

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Journal ArticleDOI
TL;DR: In this article, a sommaire de plusieurs metriques/distances of probabilite couramment utilisees par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux resultats qui se rapportent a leurs bornes.
Abstract: Le choix de metrique de probabilite est une decision tres importante lorsqu'on etudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs metriques/distances de probabilite couramment utilisees par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux resultats qui se rapportent a leurs bornes. Avoir connaissance d'autres metriques peut vous fournir avec un moyen de deriver des bornes pour une autre metrique dans un probleme applique. Le fait de prendre en consideration plusieurs metriques vous permettra d'approcher des problemes d'une maniere differente. Ainsi, nous vous demontrons que les taux de convergence peuvent dependre de facon importante sur votre choix de metrique. Il est donc important de tout considerer lorsqu'on doit choisir une metrique.

1,271 citations

Journal ArticleDOI
TL;DR: In this article, the second largest eigenvalue and spectral gap of a reversible Markov chain were derived for the random walk associated to approximate computation of the permanent. But these bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs.
Abstract: We develop bounds for the second largest eigenvalue and spectral gap of a reversible Markov chain. The bounds depend on geometric quantities such as the maximum degree, diameter and covering number of associated graphs. The bounds compare well with exact answers for a variety of simple chains and seem better than bounds derived through Cheeger-like inequalities. They offer improved rates of convergence for the random walk associated to approximate computation of the permanent.

947 citations

Journal ArticleDOI
TL;DR: For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant, deduced from a more general result about stationary finite-state reversible Markov chains.
Abstract: For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

942 citations


Cites background or methods from "Group representations in probabilit..."

  • ...See [123, 127] and [112] Chapter 3F for details on convergence to equilibrium and [110] for hitting and cover times....

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  • ...Brief accounts can be found in Letac [224, 225] and Diaconis [112] Chapter 3F, which contains an extensive annotated bibliography....

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  • ...I lack the space (and, more importantly, the knowledge) to give a worthwhile treatment here, and in any case an account which is both introductory and gets to interesting results is available in Diaconis [112]....

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  • ...jjj Cite Diaconis book [112]? Continue same paragraph: So the relaxation time is...

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  • ...3 with random transpositions benchmark, it is known from group representation methods [112, 122] which make essential use of all the eigenvalues λ̃r, not just λ̃2, that ̃̂ τ ∼ 1 2m logm as m→∞....

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References
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Book
01 Jan 1983

34,729 citations

Book
01 Jan 1983
TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

23,215 citations

Book
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.

17,939 citations

Journal ArticleDOI
TL;DR: In this article, Lindley et al. make the less restrictive assumption that such a normal, homoscedastic, linear model is appropriate after some suitable transformation has been applied to the y's.
Abstract: [Read at a RESEARCH METHODS MEETING of the SOCIETY, April 8th, 1964, Professor D. V. LINDLEY in the Chair] SUMMARY In the analysis of data it is often assumed that observations Yl, Y2, *-, Yn are independently normally distributed with constant variance and with expectations specified by a model linear in a set of parameters 0. In this paper we make the less restrictive assumption that such a normal, homoscedastic, linear model is appropriate after some suitable transformation has been applied to the y's. Inferences about the transformation and about the parameters of the linear model are made by computing the likelihood function and the relevant posterior distribution. The contributions of normality, homoscedasticity and additivity to the transformation are separated. The relation of the present methods to earlier procedures for finding transformations is discussed. The methods are illustrated with examples.

12,158 citations