Polya Urn Research Articles

Distributions of cherries for two models of trees

Null models for generating binary phylogenetic trees are useful for testing evolutionary hypotheses and reconstructing phylogenies. We consider two such null models – the Yule and uniform models – and in particular the induced distribution they generate on the number C n of cherries in the tree, where a cherry is a pair of leaves each of which is adjacent to a common ancestor. By realizing the process of cherry formation in these two models by extended Polya urn models we show that C n is asymptotically normal. We also give exact formulas for the mean and standard deviation of the C n in these two models. This allows simple statistical tests for the Yule and uniform null hypotheses.

Image segmentation and labeling using the Polya urn model

We propose a segmentation method based on Polya's (1931) urn model for contagious phenomena. A preliminary segmentation yields the initial composition of an urn representing the pixel. The resulting urns are then subjected to a modified urn sampling scheme mimicking the development of an infection to yield a segmentation of the image into homogeneous regions. This process is implemented using contagion urn processes and generalizes Polya's scheme by allowing spatial interactions. The composition of the urns is iteratively updated by assuming a spatial Markovian relationship between neighboring pixel labels. The asymptotic behavior of this process is examined and comparisons with simulated annealing and relaxation labeling are presented. Examples of the application of this scheme to the segmentation of synthetic texture images, ultra-wideband synthetic aperture radar (UWB SAR) images and magnetic resonance images (MRI) are provided.

Exponential posterior consistency via generalized Pólya urn schemes in finite semiparametric mixtures

Advances in Markov chain Monte Carlo MCMC methods now make it computationally feasible and relatively straightforward to apply the Dirichlet process prior in a wide range of Bayesian nonparametric problems. The feasibility of these methods rests heavily on the fact that the MCMC approach avoids direct sampling of the Dirichlet process and is instead based on sampling the finite-dimensional posterior which is obtained from marginalizing out the process. In application, it is the integrated posterior that is used in the Bayesian nonparametric inference, so one might wonder about its theoretical properties. This paper presents some results in this direction. In particular, we will focus on a study of the posterior’s asymptotic behavior, specifically for the problem when the data is obtained from a finite semiparametric mixture distribution. A complication in the analysis arises because the dimension for the posterior, although finite, increases with the sample size. The analysis will reveal general conditions that ensure exponential posterior consistency for a finite dimensional parameter and which can be slightly generalized to allow the unobserved nonparametric parameters to be sampled from a generalized Polya urn scheme. Several interesting examples are considered.

A Note on Variable Replacement Rates in Urn Models

(1998). A Note on Variable Replacement Rates in Urn Models. The American Mathematical Monthly: Vol. 105, No. 8, pp. 764-765.

Self organization of interacting polya urns

We introduce a simple model which shows non-trivial self organized critical properties. The model describes a system of interacting units, modelled by Polya urns, subject to perturbations and which occasionally break down. Three equivalent formulations - stochastic, quenched and deterministic - are shown to reproduce the same dynamics. Among the novel features of the model are a non-homogeneous stationary state, the presence of a non-stationary critical phase and non-trivial exponents even in mean field. We discuss simple interpretations in term of biological evolution and earthquake dynamics and we report on extensive numerical simulations in dimensions d=1,2 as well as in the random neighbors limit.

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Polya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Polya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].

On convergence of series of random variables with applications to martingale convergence and to convergence of series with orthogonal summands

The rate of convergence for an almost surely convergent series of random variables {Xn, n ≥ 1} is studied in this paper. More specifically, if Sn converges almost surely to a random variable S then the tail series is a well-defined sequence of random variable with Tn → 0 almost surely. The main result provides conditions for each of to hold for given numerical sequences . As special cases, new results are obtained when {Xn, n ≥ 1} is a martingale difference sequence and when the Xn 's are of the form , where {an, n ≥ 1} is a sequence of constants and {Yn, n ≥ 1} is an orthogonal system of exchangeable random variables. An application to the Polya urn scheme is considered. The current work generalizes and simplifies some of the recent results of Nam and Rosalsky [15]

A communication channel modeled on contagion

We introduce a binary additive communication channel with memory. The noise process of the channel is generated according to the contagion model of G. Polya (1923); our motivation is the empirical observation of Stapper et al. (1980) that defects in semiconductor memories are well described by distributions derived from Polya's urn scheme. The resulting channel is stationary but not ergodic, and it has many interesting properties. We first derive a maximum likelihood (ML) decoding algorithm for the channel; it turns out that ML decoding is equivalent to decoding a received vector onto either the closest codeword or the codeword that is farthest away, depending on whether an apparent epidemic has occurred. We next show that the Polya-contagion channel is an averaged channel in the sense of Ahlswede (1968) and others and that its capacity is zero. Finally, we consider a finite-memory version of he Polya-contagion model; this channel is (unlike the original) ergodic with a nonzero capacity that increases with increasing memory. >

Martingale Functional Central Limit Theorems for a Generalized Polya Urn

In a generalized two-color Polya urn scheme, allowing negative replacements, we use martingale techniques to obtain weak invariance principles for the urn process $(W_n)$, where $W_n$ is the number of white balls in the urn at stage $n$. The normalizing constants and the limiting Gaussian process are shown to depend on the ratio of the eigenvalues of the replacement matrix.

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

AbstractThe generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two‐parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two‐parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub‐binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.

A p�lya urn model with a continuum of colors

For a Polya urn model with a continuum of colors introduced by Blackwell and MacQueen ((1973),Ann. Statist.,2, 1152–1174), we show the joint distribution of colors aftern draws from which several properties of the urn model are derived. The similar results hold for the case where the initial distribution of colors is invariant under a finite group of transformations.

On the structure of random plane‐oriented recursive trees and their branches

AbstractThis paper is an investigation of the structural properties of random plane‐oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second‐order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.

Polya Trees and Random Distributions

Trees of Polya urns are used to generate sequences of exchangeable random variables. By a theorem of de Finetti each such sequence is a mixture of independent, identically distributed variables and the mixing measure can be viewed as a prior on distribution functions. The collection of these Polya tree priors forms a convenient conjugate family which was mentioned by Ferguson and includes the Dirichlet processes of Ferguson. Unlike Dirichlet processes, Polya tree priors can assign probability 1 to the class of continuous distributions. This property and a few others are investigated.

When are Touchpoints Limits for Generalized Polya URNS?

When are Touchpoints Limits for Generalized Polya URNS?

Reinforced Random Walks and Random Distributions

Consider a classical Polya urn process on a complete binary tree. This process generates an exchangeable sequence of random variables ${Z_n}$, with values in $[0,1]$. It is shown that the empirical distribution $^\# \{ i \leq n:{Z_i} \leq s\} /n$ converges weakly and the distribution of this limit is the same as a standard Dubins-Freedman random distribution. As an application, the variance of the first moment of these Dubins-Freedman distributions is calculated.

Reinforced random walks and random distributions

Consider a classical Polya urn process on a complete binary tree. This process generates an exchangeable sequence of random variables ${Z_n}$, with values in $[0,1]$. It is shown that the empirical distribution $^\# \{ i \leq n:{Z_i} \leq s\} /n$ converges weakly and the distribution of this limit is the same as a standard Dubins-Freedman random distribution. As an application, the variance of the first moment of these Dubins-Freedman distributions is calculated.

Exchangeable Urn Processes

If $Y_n$ is 1 or 0 depending on whether the $n$th ball drawn in a Polya urn scheme is red or not, then the variables $Y_1, Y_2,\ldots$ are exchangeable. It is shown for a generalized class of urn models that no other scheme gives rise to exchangeable variables unless the $Y_n$ are either independent and identically distributed, or deterministic (that is, all of the $Y_n$'s have the same value with probability 1).

Random Walks of Ordered Elements with Applications

Abstract In this partly expository paper a simple random walk model is introduced for generating sequences of ordered elements. The model is related to Polya's urn scheme. An alternative description of the walk leads to conditionally independent steps. A basic theorem for the walk is proved and applications are given. Among other things, recursion formulas for moments of order statistics are studied and connections with exceedances and distribution-free tolerance intervals are indicated.

Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles

It has recently been shown that the Ewens sampling formula may be generated by a Polya-like urn model. A genealogical proof of this result equates the labelling of balls in the urn to the partition by age of alleles in the sample. This urn construction is shown to be equivalent to the construction of Kingman ( Proc. Roy. Soc. London Ser. A 361 (1978), 1–20) using a Poisson-Dirichlet “paintbox” and as a consequence, the partition by ages is seen to be equivalent to the size biased permutation of the Poisson-Dirichlet distribution. This approach unifies and extends many results on ages of alleles, the Polya urn, and the Poisson-Dirichlet distribution. Furthermore the Ewens sampling formula is characterized as being the only partition structure which may be generated by an urn-like mechanism.

A note on plurality distortion in large committees

Simple plurality voting allows the somewhat surprising possibility that a majority of a committee or an electorate may prefer one of the defeated alternatives to the plurality winner. This paradox is distinct from Condorcet's paradox and has received less attention in the literature. Recently, however, the frequency of the Borda effect anticipated under different cultures has been computed and some empirical results have also been reported. Here results are obtained for both impartial and partial cultures using a class of Polya urn models. Emphasis is on asymptotic investigations and results valid for large committees.