Urn Process Research Articles

Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\in \\mathbb {N}$$\\end{document} be the number of urns/colors. At each time n, we distribute σn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _n$$\\end{document} balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} assuming some monotonicity and growth condition. The class R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} includes convex functions and the classical case f(x)=xα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x)=x^{\\alpha }$$\\end{document}, α>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >1$$\\end{document}. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.

Estimation of reinforced urn processes under left-truncation and right-censoring

We propose a non-parametric estimator for bivariate left-truncated and right-censored observations that combines the expectation-maximization algorithm and the reinforced urn process. The resulting expectation-reinforcement algorithm allows for the inclusion of experts' knowledge in the form of a prior distribution, thus belonging to the class of Bayesian models. This can be relevant in applications where the data is incomplete, due to biases in the sampling process, as in the case of left-truncation and right-censoring. With this new approach, the distribution of the truncation variables is also recovered, granting further insight into those biases, and playing an important role in applications like prevalent cohort studies. The estimators are tested numerically using artificial and empirical datasets, and compared with other methodologies such as copula models and the Kaplan-Meier estimator.

International trade and technological competition in markets with dynamic increasing returns

We build a simple international trade model to study how the interaction between imperfect market selection and technological learning jointly shapes trade patterns as well as firm and industry dynamics. The model features two countries populated by firms heterogeneous in their productivity and size. Market selection is driven by a finite pairwise Pólya urn process, while a geometric random walk characterizes the idiosyncratic cumulative firm learning process. We show that the model is able to jointly reproduce a wide ensemble of stylized facts concerning intra-industry trade, industry and firm dynamics as well as realistic dynamics of export and productivity leadership at the country level. A series of sensitivity exercises on the degree of trade openness and the tightness of the market selection process allow us to draw interesting policy insights on concentration, volatility and the dynamics of international leadership in both export shares and aggregate productivity.

Infinite-color randomly reinforced urns with dominant colors

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Under fairly mild assumptions on the expected reinforcement, we show that the predictive and the empirical distributions converge almost surely (a.s.) in total variation to the same random probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$ a.s., where $\mathcal{D}$ denotes the set of dominant colors for which the expected reinforcement is maximum. In the general case, the predictive probabilities and the empirical frequencies of any $\delta$-neighborhood of $\mathcal{D}$ converge a.s. to one. That is, although non-dominant colors continue to be regularly observed, their distance to $\mathcal{D}$ converges in probability to zero. We refine the above results with rates of convergence. We further hint potential applications of dominant P\'olya sequences in randomized clinical trials and species sampling, and use our central limit results for Bayesian inference.

From the multiterm urn model to the self-exciting negative binomial distribution and Hawkes processes.

This paper considers a multiterm urn process that has a correlation in the same term and temporal correlation. .The objective is to clarify the relationship between the urn model and the Hawkes process. Correlation in the same term is represented by the Pólya urn model and the temporal correlation is incorporated by introducing the conditional initial condition. In the double scaling limit of this urn process, the self-exciting negative binomial distribution process, which is a marked Hawkes process, is obtained. In the standard continuous limit, this process becomes the Hawkes process, which has no correlation in the same term. The difference is the variance of the intensity function in that the phase transition from the steady to the nonsteady state can be observed. The critical point, at which the power-law distribution is obtained, is the same for the Hawkes and the urn processes. These two processes are used to analyze empirical data of financial default to estimate the parameters of the model. For the default portfolio, the results produced by the urn process are superior to those obtained with the Hawkes process and confirm self-excitation.

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Joint and survivor annuity valuation with a bivariate reinforced urn process

We introduce a novel way of modeling the dependence of coupled lifetimes, for the pricing of joint and survivor annuities. Using a well-known Canadian data set, our results are analyzed and compared with the existing literature, mainly relying on copulas. Based on urn processes and a one-factor construction, the proposed model is able to improve its performances over time, in line with the machine learning paradigm, and it also allows for the use of experts' judgements, to complement the empirical data.

Gaussian process approximations for multicolor Pólya urn models

AbstractMotivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

International Trade and Technological Competition in Markets with Dynamic Increasing Returns

We build a simple dynamic model to study the effects of technological learning, market selection and international competition in the determination of export flows and market shares. The model features two countries populated by firms with heterogeneous productivity levels and sales. Market selection in each country is driven by a finite pairwise Polya urn process. We show that market selection leads either to a national or to an international monopoly in presence of a static distribution of firm productivity levels. We then incorporate firm learning and entry-exit in the model and we show that the market structure does not converge to a monopoly. In addition, we show that the extended model is able to jointly reproduce a wide ensemble of stylized facts concerning intra-industry trade, industry and firm dynamics.

Bayesian Allocation Model: Marginal Likelihood-Based Model Selection for Count Tensors

In this article, we introduce a dynamic generative model, the Bayesian allocation model (BAM), for modeling count data. BAM covers various probabilistic nonnegative tensor factorization (NTF) and topic models under one general framework. In BAM, allocations are made using a Bayesian network, whose conditional probability tables can be integrated out analytically. We show that, when allocations are viewed as sequential, the resulting marginal process is a special type of Polya urn process, which we name as Polya-Bayes process, an integer valued self-reinforcing process. Exploiting the Polya urn construction, we develop a novel sequential Monte Carlo (SMC) algorithm for marginal likelihood estimation in BAM, leading to a unified scoring method for discrete variable Bayesian networks with hidden nodes, including various NTF and topic models. The SMC estimator for marginal likelihood has the remarkable property of being unbiased in contrast to variational algorithms which are generally biased. We also demonstrate how our novel SMC-based likelihood estimation can be integrated within a Markov chain Monte Carlo algorithm for a principled and correct (in terms of respecting the true posterior distribution) Bayesian model selection and hyperparameter estimation for BAM. We provide several numerical examples, both on artificial and real datasets, that demonstrate the performance of the algorithms for various data regimes.

Herding, Learning and Incentives for Online Reviews

We investigate the role of consumer herding and learning on the design of incentives for online customer reviews. Herding occurs when consumers are drawn to a product that appears to be popular because it has garnered a large number of reviews. Learning occurs when consumers infer product quality from reviews. We evaluate and compare three incentive policies. The first announces an incentive to all customers before purchase, the second offers an incentive after purchase, and the third rewards buyers only if they write positive, possibly fake, reviews. We use a generalized Polya urn process to model the evolution of reviews. The expected value of the resulting aggregate demand has the form of the Gompertz function. We obtain conditions under which each type of incentive is profitable, and preferred by a seller to the other incentives for reviews. The results imply that sellers should use different incentives policies depending on the quality and profit margin of a product. A pre-purchase incentive is the most profitable when product quality and profit margin are both high; an incentive offered to buyers after obtaining voluntary reviews is the most profitable when product quality is high and profit margin is low; and an incentive for only positive reviews is the most profitable when product quality and profit margin are both low. E-commerce platforms that limit their sellers to using post-purchase incentives might be more effective in curbing fake reviews if they also allow sellers to announce pre-purchase incentives to all customers.

Online popularity of destinations in Australia: An application of Polya Urn process to search engine data

The online search engine is a key source of tourism information for trip planning. In the internet, experience and feedback shared by previous tourists can be obtained with increasing ease, resulting in a feedback mechanism of interpersonal influence featured in the online peer-to-peer tourism information exchange. Such interpersonal influence may lead to a unique popularity distribution among destinations. The information exchange process is also affected by the destination choice set, which is determined by the geography of the destination countries. This study mathematically characterised the e-word-of-mouth effect in travel and tourism on the pattern formation in online destination. To achieve this, the Polya Urn process is invoked to model the clustering of destination preference and the distribution of destination popularity in an online environment. Google search volume on various destinations in Australia and France from five tourism inbound countries are used. Results show that the theoretical distribution of the Polya Urn process fits the distribution of search volume on Australian destinations, highlighting the geographical uniqueness of Australia as a closed system that possess a unique online popularity growth process and distribution.

A martingale approach for Pólya urn processes

This paper is devoted to a direct martingale approach for Polya urn models asymptotic behaviour. A Polya process is said to be small when the ratio of its remplacement matrix eigenvalues is less than or equal to 1/2, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urns processes. We also provide new almost sure properties for small urns processes.

Joint and Survivor Annuity Valuation with a Bivariate Reinforced Urn Process

We introduce a novel way of modeling the dependence of coupled lifetimes, for the pricing of joint and survivor annuities. Using a well-known Canadian data set, our results are analysed and compared with the existing literature, mainly relying on copulas. Based on urn processes and a one-factor construction, the proposed model is able to improve its performances over time, in line with the machine learning paradigm, and it also allows for the use of experts’ judgements, to complement the empirical data.

Estimation for Uni-variate and Bi-variate Reinforced Urn Processes Under Left-Truncation and Right-Censoring

Reinforced Urn Processes (RUPs) represent a flexible class of Bayesian nonparametric models suitable for dealing with possibly right-censored and left-truncated observations. A reliable estimation of their hyper-parameters is however missing in the literature. We therefore propose an extension of the Expectation-Maximization (EM) algorithm for RUPs, both in the univariate and the bivariate case. Furthermore, a new methodology combining EM and the prior elicitation mechanism of RUPs is developed: the Expectation-Reinforcement algorithm. Numerical results showing the performance of both algorithms are presented for several analytical examples as well as for a large dataset of Canadian annuities.

Revelation, Decision and Fascination

We model the sequential decisions made by receivers in response to multiple signals. In the first part, we model this in a special setting where the signals in the system are generated by either truth-telling or lying. We classify the signals as binary high or low, and study the information herding behavior. Then we study the decision-making in response to signals that cannot be simply classified as binary high or low. Last, we apply the Polya urn process to model the dynamics of public announcements made by the receivers in a more general setting, and discuss how senders can design their strategies to take advantage of public announcements and further increase their benefits. This research quantitatively examines people’s reactions to signals, and can be generalized and adapted to cases like the design of game shows, advertising, biding, and customer review management.

Preferential attachment when stable

AbstractWe study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha$th power $(\alpha >1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower-tail large deviation principle (LDP) for $L = \sum_{i=1}^{n} ({S_i^2}/{i^2})$, where $\{S_n \colon n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

Curing Epidemics on Networks Using a Polya Contagion Model

We study the curing of epidemics of a network contagion, which is modelled using a variation of the classical Polya urn process that takes into account spatial infection among neighbouring nodes. We introduce several quantities for measuring the overall infection in the network and use them to formulate an optimal control problem for minimizing the average infection rate using limited curing resources. We prove the feasibility of this problem under high curing budgets by deriving conservative lower bounds on the amount of curing per node that turn our measures of network infection into supermartingales. We also provide a provably convergent gradient descent algorithm to find the allocation of curing under limited budgets. Motivated by the fact that this strategy is computationally expensive, we design a suit of heuristic methods that are locally implementable and nearly as effective. Extensive simulations run on large-scale networks demonstrate the effectiveness of our proposed strategies.

An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages

We propose an alternative approach to the modeling of the positive dependence between the probability of default and the loss given default in a portfolio of exposures, using a bivariate urn process. The model combines the power of Bayesian nonparametrics and statistical learning, allowing for the elicitation and the exploitation of experts’ judgements, and for the constant update of this information over time, every time new data are available. A real-world application on mortgages is described using the Single Family Loan-Level Dataset by Freddie Mac.

Spatial Distribution of Tourism Activities: A Polya Urn Process Model of Rank-Size Distribution

The power law is considered one of the most enduring regularities in human geography. This article aims to develop an understanding of the circumstances that may result in the power law distribution in the geography of tourism activities. The finite Polya urn process is adopted as a device to model the preferential attachment process in the flow of tourists. The model generates a rank-size distribution of tourism regions along with intuitively appealing parameters. Empirically examined using two independent sets of Australian inbound and outbound tourism data, results show that the rank-size distribution emerging from the finite Polya urn process is a superior fit to the conventional power law curve. This rank-size distribution (termed the Polya urn process model of visitor distribution) is compatible with tourist behaviors such as habit persistence and word-of-mouth effects, and can be adopted by tourism modelers to predict and efficiently summarize the spatiality of tourism.