The Experts below are selected from a list of 213 Experts worldwide ranked by ideXlab platform

John H Halton - One of the best experts on this subject based on the ideXlab platform.

  • Sigma Algebra theorems
    Monte Carlo Methods and Applications, 2008
    Co-Authors: John H Halton
    Abstract:

    In reviewing the foundations of probability, to establish the underpinnings of the Monte Carlo method, I found some basic results for which I could not find a simple, clear, and concise derivation in the literature. This paper is an attempt to present a sufficient collection of such results. The presentation is pitched at the level of Monte Carlo practitioners, rather than mathematical probabilists.

  • quasi probability why quasi monte carlo methods are statistically valid and how their errors can be estimated statistically
    Monte Carlo Methods and Applications, 2005
    Co-Authors: John H Halton
    Abstract:

    The classical model of probability theory, due principally to Kolmogorov, defines probability as a totally-one measure on a Sigma-Algebra of subsets (events) of a given set (the sample space), and random variables as real-valued functions on the sample space, such that the inverse images of all Borel sets are events. From this model, all the results of probability theory are derived. However, the assertion that any given concrete situation is subject to probability theory is a scientific hypothesis verifiable only experimentally, by appropriate sampling, and never totally certain. Furthermore, classical probability theory allows for the possibility of "outliers"—sampled values which are misleading. In particular, Kolmogorov's Strong Law of Large Numbers asserts that, if, as is usually the case, a random variable has a finite expectation (its integral over the sample space), then the average value of N independently sampled values of this function converges to the expectation with probability 1, as N tends to infinity. This implies that there may be sample sequences (belonging to a set of probability 0) for which this convergence does not occur; these are the "outliers".

Zakhar Kabluchko - One of the best experts on this subject based on the ideXlab platform.

  • a functional central limit theorem for branching random walks almost sure weak convergence and applications to random trees
    Annals of Applied Probability, 2016
    Co-Authors: Rudolf Grubel, Zakhar Kabluchko
    Abstract:

    Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_{n}(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process \[D_{n}(u):=m^{\frac{1}{2}n}(W_{\infty}(\frac{u}{\sqrt{n}})-W_{n}(\frac{u}{\sqrt{n}}))\] converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result, we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Random Structures Algorithms46 (2015) 346–361], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger’s theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that \[\mathcal{L}\{\sqrt{\frac{n}{2\log n}}(\operatorname{EPL}_{\infty}-\frac{\operatorname{EPL}_{n}-2n\log n}{n})\Big |\mathcal{G}_{n}\}\overset{\mathrm{a.s.w.}}{\underset{n\to\infty}\longrightarrow}\{\omega \mapsto\mathcal{N}_{0,1}\},\] where $\operatorname{EPL}_{n}$ is the external path length of a binary search tree $X_{n}$ with $n$ vertices, $\operatorname{EPL}_{\infty}$ is the limit of the Regnier martingale and $\mathcal{L}\{\cdot |\mathcal{G}_{n}\}$ denotes the conditional distribution w.r.t. the $\Sigma$-Algebra $\mathcal{G}_{n}$ generated by $X_{1},\ldots,X_{n}$. Almost sure weak convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton–Watson processes and the Polya urn.

  • a functional central limit theorem for branching random walks almost sure weak convergence and applications to random trees
    arXiv: Probability, 2014
    Co-Authors: Rudolf Grubel, Zakhar Kabluchko
    Abstract:

    Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger's theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that $$ L\left\{\sqrt{\frac{n}{2\log n}} \left(EPL_{\infty} - \frac{EPL_n-2n\log n}{n}\right)\Bigg | G_{n}\right\} \to \{\omega\mapsto N_{0,1}\}, \quad \text{a.s.w.},$$ where $EPL_n$ is the external path length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\infty}$ is the limit of the R\'egnier martingale, and $L(\,\cdot\, |G_n)$ denotes the conditional distribution w.r.t. the $\Sigma$-Algebra $G_n$ generated by $X_1,\ldots,X_n$. A.s.w. convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton-Watson processes and the P\'olya urn.

Rudolf Grubel - One of the best experts on this subject based on the ideXlab platform.

  • a functional central limit theorem for branching random walks almost sure weak convergence and applications to random trees
    Annals of Applied Probability, 2016
    Co-Authors: Rudolf Grubel, Zakhar Kabluchko
    Abstract:

    Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_{n}(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process \[D_{n}(u):=m^{\frac{1}{2}n}(W_{\infty}(\frac{u}{\sqrt{n}})-W_{n}(\frac{u}{\sqrt{n}}))\] converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result, we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Random Structures Algorithms46 (2015) 346–361], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger’s theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that \[\mathcal{L}\{\sqrt{\frac{n}{2\log n}}(\operatorname{EPL}_{\infty}-\frac{\operatorname{EPL}_{n}-2n\log n}{n})\Big |\mathcal{G}_{n}\}\overset{\mathrm{a.s.w.}}{\underset{n\to\infty}\longrightarrow}\{\omega \mapsto\mathcal{N}_{0,1}\},\] where $\operatorname{EPL}_{n}$ is the external path length of a binary search tree $X_{n}$ with $n$ vertices, $\operatorname{EPL}_{\infty}$ is the limit of the Regnier martingale and $\mathcal{L}\{\cdot |\mathcal{G}_{n}\}$ denotes the conditional distribution w.r.t. the $\Sigma$-Algebra $\mathcal{G}_{n}$ generated by $X_{1},\ldots,X_{n}$. Almost sure weak convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton–Watson processes and the Polya urn.

  • a functional central limit theorem for branching random walks almost sure weak convergence and applications to random trees
    arXiv: Probability, 2014
    Co-Authors: Rudolf Grubel, Zakhar Kabluchko
    Abstract:

    Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger's theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that $$ L\left\{\sqrt{\frac{n}{2\log n}} \left(EPL_{\infty} - \frac{EPL_n-2n\log n}{n}\right)\Bigg | G_{n}\right\} \to \{\omega\mapsto N_{0,1}\}, \quad \text{a.s.w.},$$ where $EPL_n$ is the external path length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\infty}$ is the limit of the R\'egnier martingale, and $L(\,\cdot\, |G_n)$ denotes the conditional distribution w.r.t. the $\Sigma$-Algebra $G_n$ generated by $X_1,\ldots,X_n$. A.s.w. convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton-Watson processes and the P\'olya urn.

Daniel J Rudolph - One of the best experts on this subject based on the ideXlab platform.

  • pointwise and l sp 1 mixing relative to a sub Sigma Algebra
    Illinois Journal of Mathematics, 2004
    Co-Authors: Daniel J Rudolph
    Abstract:

    We consider two natural definitions for the no- tion of a dynamical system being mixing relative to an in- variant sub -AlgebraH. Both concern the convergence of |E(f·g T n |H) E(f|H)E(g T n |H)|! 0 as|n| ! 1 for appropriate f and g. The weaker condition asks for convergence in L 1 and the stronger for convergence a.e. We will see that these are dierent conditions. Our goal is to show that both these notions are robust. As is quite standard we show that one need only consider g = f and E(f|H) = 0, and in this case |E(f · f T n |H)| ! 0. We will see rather easily that for L 1 convergence it is enough to check an L 2 -dense family. Our major result will be to show the same is true for pointwise convergence making this a verifiable condition. As an application we will see that if T is mixing then for any ergodic S, S◊ T is relatively mixing with respect to the first coordinate sub -Algebra in the pointwise sense.

Joan Bruna - One of the best experts on this subject based on the ideXlab platform.

  • on the equivalence between graph isomorphism testing and function approximation with gnns
    arXiv: Learning, 2019
    Co-Authors: Zhengdao Chen, Soledad Villar, Lei Chen, Joan Bruna
    Abstract:

    Graph neural networks (GNNs) have achieved lots of success on graph-structured data. In the light of this, there has been increasing interest in studying their representation power. One line of work focuses on the universal approximation of permutation-invariant functions by certain classes of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism tests. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the representation power of GNNs with the language of Sigma-Algebra, which incorporates both viewpoints. Using this framework, we compare the expressive power of different classes of GNNs as well as other methods on graphs. In particular, we prove that order-2 Graph G-invariant networks fail to distinguish non-isomorphic regular graphs with the same degree. We then extend them to a new architecture, Ring-GNNs, which succeeds on distinguishing these graphs and provides improvements on real-world social network datasets.

  • on the equivalence between graph isomorphism testing and function approximation with gnns
    Neural Information Processing Systems, 2019
    Co-Authors: Zhengdao Chen, Soledad Villar, Lei Chen, Joan Bruna
    Abstract:

    Graph neural networks (GNNs) have achieved lots of success on graph-structured data. In light of this, there has been increasing interest in studying their representation power. One line of work focuses on the universal approximation of permutation-invariant functions by certain classes of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism tests. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the representation power of GNNs with the language of Sigma-Algebra, which incorporates both viewpoints. Using this framework, we compare the expressive power of different classes of GNNs as well as other methods on graphs. In particular, we prove that order-2 Graph G-invariant networks fail to distinguish non-isomorphic regular graphs with the same degree. We then extend them to a new architecture, Ring-GNN, which succeeds in distinguishing these graphs as well as for tasks on real-world datasets.