Basic Definitions: Indexed Collections and Random Functions
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On the Mean Speed of Convergence of Empirical and Occupation Measures in Wasserstein Distance Emmanuel Boissard, Thibaut Le Gouic
On the mean speed of convergence of empirical and occupation measures in Wasserstein distance Emmanuel Boissard, Thibaut Le Gouic To cite this version: Emmanuel Boissard, Thibaut Le Gouic. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2014, 10.1214/12-AIHP517. hal-01291298 HAL Id: hal-01291298 https://hal.archives-ouvertes.fr/hal-01291298 Submitted on 8 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON THE MEAN SPEED OF CONVERGENCE OF EMPIRICAL AND OCCUPATION MEASURES IN WASSERSTEIN DISTANCE EMMANUEL BOISSARD AND THIBAUT LE GOUIC Abstract. In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures. -
1.5. Set Functions and Properties. Let Л Be Any Set of Subsets of E
1.5. Set functions and properties. Let A be any set of subsets of E containing the empty set ∅. A set function is a function µ : A → [0, ∞] with µ(∅)=0. Let µ be a set function. Say that µ is increasing if, for all A, B ∈ A with A ⊆ B, µ(A) ≤ µ(B). Say that µ is additive if, for all disjoint sets A, B ∈ A with A ∪ B ∈ A, µ(A ∪ B)= µ(A)+ µ(B). Say that µ is countably additive if, for all sequences of disjoint sets (An : n ∈ N) in A with n An ∈ A, S µ An = µ(An). n ! n [ X Say that µ is countably subadditive if, for all sequences (An : n ∈ N) in A with n An ∈ A, S µ An ≤ µ(An). n ! n [ X 1.6. Construction of measures. Let A be a set of subsets of E. Say that A is a ring on E if ∅ ∈ A and, for all A, B ∈ A, B \ A ∈ A, A ∪ B ∈ A. Say that A is an algebra on E if ∅ ∈ A and, for all A, B ∈ A, Ac ∈ A, A ∪ B ∈ A. Theorem 1.6.1 (Carath´eodory’s extension theorem). Let A be a ring of subsets of E and let µ : A → [0, ∞] be a countably additive set function. Then µ extends to a measure on the σ-algebra generated by A. Proof. For any B ⊆ E, define the outer measure ∗ µ (B) = inf µ(An) n X where the infimum is taken over all sequences (An : n ∈ N) in A such that B ⊆ n An and is taken to be ∞ if there is no such sequence. -
A Bayesian Level Set Method for Geometric Inverse Problems
Interfaces and Free Boundaries 18 (2016), 181–217 DOI 10.4171/IFB/362 A Bayesian level set method for geometric inverse problems MARCO A. IGLESIAS School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK E-mail: [email protected] YULONG LU Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK E-mail: [email protected] ANDREW M. STUART Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK E-mail: [email protected] [Received 3 April 2015 and in revised form 10 February 2016] We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data, and may be used to not only estimate interface locations, but quantify uncertainty in them; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC methodology applied to the level set function – no explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications. -
Generalized Stochastic Processes with Independent Values
GENERALIZED STOCHASTIC PROCESSES WITH INDEPENDENT VALUES K. URBANIK UNIVERSITY OF WROCFAW AND MATHEMATICAL INSTITUTE POLISH ACADEMY OF SCIENCES Let O denote the space of all infinitely differentiable real-valued functions defined on the real line and vanishing outside a compact set. The support of a function so E X, that is, the closure of the set {x: s(x) F 0} will be denoted by s(,o). We shall consider 3D as a topological space with the topology introduced by Schwartz (section 1, chapter 3 in [9]). Any continuous linear functional on this space is a Schwartz distribution, a generalized function. Let us consider the space O of all real-valued random variables defined on the same sample space. The distance between two random variables X and Y will be the Frechet distance p(X, Y) = E[X - Yl/(l + IX - YI), that is, the distance induced by the convergence in probability. Any C-valued continuous linear functional defined on a is called a generalized stochastic process. Every ordinary stochastic process X(t) for which almost all realizations are locally integrable may be considered as a generalized stochastic process. In fact, we make the generalized process T((p) = f X(t),p(t) dt, with sp E 1, correspond to the process X(t). Further, every generalized stochastic process is differentiable. The derivative is given by the formula T'(w) = - T(p'). The theory of generalized stochastic processes was developed by Gelfand [4] and It6 [6]. The method of representation of generalized stochastic processes by means of sequences of ordinary processes was considered in [10]. -
Pdf File of Second Edition, January 2018
Probability on Graphs Random Processes on Graphs and Lattices Second Edition, 2018 GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge copyright Geoffrey Grimmett Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 56 Figures copyright Geoffrey Grimmett Contents Preface ix 1 Random Walks on Graphs 1 1.1 Random Walks and Reversible Markov Chains 1 1.2 Electrical Networks 3 1.3 FlowsandEnergy 8 1.4 RecurrenceandResistance 11 1.5 Polya's Theorem 14 1.6 GraphTheory 16 1.7 Exercises 18 2 Uniform Spanning Tree 21 2.1 De®nition 21 2.2 Wilson's Algorithm 23 2.3 Weak Limits on Lattices 28 2.4 Uniform Forest 31 2.5 Schramm±LownerEvolutionsÈ 32 2.6 Exercises 36 3 Percolation and Self-Avoiding Walks 39 3.1 PercolationandPhaseTransition 39 3.2 Self-Avoiding Walks 42 3.3 ConnectiveConstantoftheHexagonalLattice 45 3.4 CoupledPercolation 53 3.5 Oriented Percolation 53 3.6 Exercises 56 4 Association and In¯uence 59 4.1 Holley Inequality 59 4.2 FKG Inequality 62 4.3 BK Inequalitycopyright Geoffrey Grimmett63 vi Contents 4.4 HoeffdingInequality 65 4.5 In¯uenceforProductMeasures 67 4.6 ProofsofIn¯uenceTheorems 72 4.7 Russo'sFormulaandSharpThresholds 80 4.8 Exercises 83 5 Further Percolation 86 5.1 Subcritical Phase 86 5.2 Supercritical Phase 90 5.3 UniquenessoftheIn®niteCluster 96 5.4 Phase Transition 99 5.5 OpenPathsinAnnuli 103 5.6 The Critical Probability in Two Dimensions 107 -
Mean Field Methods for Classification with Gaussian Processes
Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi cation models with Gaussian processes. In contrast to previous ap proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997). -
Old Notes from Warwick, Part 1
Measure Theory, MA 359 Handout 1 Valeriy Slastikov Autumn, 2005 1 Measure theory 1.1 General construction of Lebesgue measure In this section we will do the general construction of σ-additive complete measure by extending initial σ-additive measure on a semi-ring to a measure on σ-algebra generated by this semi-ring and then completing this measure by adding to the σ-algebra all the null sets. This section provides you with the essentials of the construction and make some parallels with the construction on the plane. Throughout these section we will deal with some collection of sets whose elements are subsets of some fixed abstract set X. It is not necessary to assume any topology on X but for simplicity you may imagine X = Rn. We start with some important definitions: Definition 1.1 A nonempty collection of sets S is a semi-ring if 1. Empty set ? 2 S; 2. If A 2 S; B 2 S then A \ B 2 S; n 3. If A 2 S; A ⊃ A1 2 S then A = [k=1Ak, where Ak 2 S for all 1 ≤ k ≤ n and Ak are disjoint sets. If the set X 2 S then S is called semi-algebra, the set X is called a unit of the collection of sets S. Example 1.1 The collection S of intervals [a; b) for all a; b 2 R form a semi-ring since 1. empty set ? = [a; a) 2 S; 2. if A 2 S and B 2 S then A = [a; b) and B = [c; d). -
Probability on Graphs Random Processes on Graphs and Lattices
Probability on Graphs Random Processes on Graphs and Lattices GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 44 Figures c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Contents Preface ix 1 Random walks on graphs 1 1.1 RandomwalksandreversibleMarkovchains 1 1.2 Electrical networks 3 1.3 Flowsandenergy 8 1.4 Recurrenceandresistance 11 1.5 Polya's theorem 14 1.6 Graphtheory 16 1.7 Exercises 18 2 Uniform spanning tree 21 2.1 De®nition 21 2.2 Wilson's algorithm 23 2.3 Weak limits on lattices 28 2.4 Uniform forest 31 2.5 Schramm±LownerevolutionsÈ 32 2.6 Exercises 37 3 Percolation and self-avoiding walk 39 3.1 Percolationandphasetransition 39 3.2 Self-avoiding walks 42 3.3 Coupledpercolation 45 3.4 Orientedpercolation 45 3.5 Exercises 48 4 Association and in¯uence 50 4.1 Holley inequality 50 4.2 FKGinequality 53 4.3 BK inequality 54 4.4 Hoeffdinginequality 56 c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 vi Contents 4.5 In¯uenceforproductmeasures 58 4.6 Proofsofin¯uencetheorems 63 4.7 Russo'sformulaandsharpthresholds 75 4.8 Exercises 78 5 Further percolation 81 5.1 Subcritical phase 81 5.2 Supercritical phase 86 5.3 Uniquenessofthein®nitecluster 92 5.4 Phase transition 95 5.5 Openpathsinannuli 99 5.6 The critical probability in two dimensions 103 5.7 Cardy's formula 110 5.8 The -
Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography
Recent advances in percolation theory and its applications Abbas Ali Saberi aDepartment of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran Abstract Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model. Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, di- mensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive per- colation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. In the past decade, after the invention of stochastic L¨ownerevolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. -
A Representation for Functionals of Superprocesses Via Particle Picture Raisa E
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER Stochastic Processes and their Applications 64 (1996) 173-186 A representation for functionals of superprocesses via particle picture Raisa E. Feldman *,‘, Srikanth K. Iyer ‘,* Department of Statistics and Applied Probability, University oj’ Cull~ornia at Santa Barbara, CA 93/06-31/O, USA, and Department of Industrial Enyineeriny and Management. Technion - Israel Institute of’ Technology, Israel Received October 1995; revised May 1996 Abstract A superprocess is a measure valued process arising as the limiting density of an infinite col- lection of particles undergoing branching and diffusion. It can also be defined as a measure valued Markov process with a specified semigroup. Using the latter definition and explicit mo- ment calculations, Dynkin (1988) built multiple integrals for the superprocess. We show that the multiple integrals of the superprocess defined by Dynkin arise as weak limits of linear additive functionals built on the particle system. Keywords: Superprocesses; Additive functionals; Particle system; Multiple integrals; Intersection local time AMS c.lassijication: primary 60555; 60H05; 60580; secondary 60F05; 6OG57; 60Fl7 1. Introduction We study a class of functionals of a superprocess that can be identified as the multiple Wiener-It6 integrals of this measure valued process. More precisely, our objective is to study these via the underlying branching particle system, thus providing means of thinking about the functionals in terms of more simple, intuitive and visualizable objects. This way of looking at multiple integrals of a stochastic process has its roots in the previous work of Adler and Epstein (=Feldman) (1987), Epstein (1989), Adler ( 1989) Feldman and Rachev (1993), Feldman and Krishnakumar ( 1994). -
Markov Random Fields and Stochastic Image Models
Markov Random Fields and Stochastic Image Models Charles A. Bouman School of Electrical and Computer Engineering Purdue University Phone: (317) 494-0340 Fax: (317) 494-3358 email [email protected] Available from: http://dynamo.ecn.purdue.edu/»bouman/ Tutorial Presented at: 1995 IEEE International Conference on Image Processing 23-26 October 1995 Washington, D.C. Special thanks to: Ken Sauer Suhail Saquib Department of Electrical School of Electrical and Computer Engineering Engineering University of Notre Dame Purdue University 1 Overview of Topics 1. Introduction (b) Non-Gaussian MRF's 2. The Bayesian Approach i. Quadratic functions ii. Non-Convex functions 3. Discrete Models iii. Continuous MAP estimation (a) Markov Chains iv. Convex functions (b) Markov Random Fields (MRF) (c) Parameter Estimation (c) Simulation i. Estimation of σ (d) Parameter estimation ii. Estimation of T and p parameters 4. Application of MRF's to Segmentation 6. Application to Tomography (a) The Model (a) Tomographic system and data models (b) Bayesian Estimation (b) MAP Optimization (c) MAP Optimization (c) Parameter estimation (d) Parameter Estimation 7. Multiscale Stochastic Models (e) Other Approaches (a) Continuous models 5. Continuous Models (b) Discrete models (a) Gaussian Random Process Models 8. High Level Image Models i. Autoregressive (AR) models ii. Simultaneous AR (SAR) models iii. Gaussian MRF's iv. Generalization to 2-D 2 References in Statistical Image Modeling 1. Overview references [100, 89, 50, 54, 162, 4, 44] 4. Simulation and Stochastic Optimization Methods [118, 80, 129, 100, 68, 141, 61, 76, 62, 63] 2. Type of Random Field Model 5. Computational Methods used with MRF Models (a) Discrete Models i. -
Random Walk in Random Scenery (RWRS)
IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 53–65 c Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000077 Random walk in random scenery: A survey of some recent results Frank den Hollander1,2,* and Jeffrey E. Steif 3,† Leiden University & EURANDOM and Chalmers University of Technology Abstract. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On Zd, d ≥ 1, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by Z, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions. 1. Introduction Random walk in random scenery is a family of stationary random processes ex- hibiting amazingly rich behavior. We will survey some of the results that have been obtained in recent years and list some open questions. Mike Keane has made funda- mental contributions to this topic. As close colleagues it has been a great pleasure to work with him. We begin by defining the object of our study. Fix an integer d ≥ 1.