Generalized Polya Urn Research Articles

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.

Optimising topic coherence with Weighted P[formula omitted]lya Urn scheme

Topic models have been widely used to mine hidden topics from documents. However, one limitation of such topic models is that they are prone to generate incoherent topics. To address this limitation, many approaches have been proposed to incorporate the prior knowledge of word semantic relatedness into the topic inference process. One example is the Generalized Po´lya Urn (GPU) scheme. However, GPU-based topic models often require sophisticated algorithms to acquire domain-specific knowledge from data. Moreover, prior knowledge is incorporated into the topic inference process without considering its impact on the intermediate topic sampling results. In this paper, we propose a novel Weighted Po´lya Urn scheme and incorporate it into Latent Dirichlet Allocation framework to build the self-enhancement topic model and generate coherent topics. In specific, semantic prior knowledge based on word embedding is employed to measure the semantic coherence of a word to different topics, which is incorporated into the Weighted Po´lya Urn scheme. Moreover, semantic coherence is updated dynamically based on the semantic similarity between a word and the representative words in different topics. Experiments have been conducted on seven public corpora from different domains to evaluate the effectiveness of the proposed approach. Experimental results show that compared to the state-of-the-art baselines, the proposed approach can generate more coherent topics.

Multi-label dataless text classification with topic modeling

Manually labeling documents is tedious and expensive, but it is essential for training a traditional text classifier. In recent years, a few dataless text classification techniques have been proposed to address this problem. However, existing works mainly center on single-label classification problems, that is, each document is restricted to belonging to a single category. In this paper, we propose a novel Seed-guided Multi-label Topic Model, named SMTM. With a few seed words relevant to each category, SMTM conducts multi-label classification for a collection of documents without any labeled document. In SMTM, each category is associated with a single category-topic which covers the meaning of the category. To accommodate with multi-label documents, we explicitly model the category sparsity in SMTM by using spike and slab prior and weak smoothing prior. That is, without using any threshold tuning, SMTM automatically selects the relevant categories for each document. To incorporate the supervision of the seed words, we propose a seed-guided biased GPU (i.e., generalized Polya urn) sampling procedure to guide the topic inference of SMTM. Experiments on two public datasets show that SMTM achieves better classification accuracy than state-of-the-art alternatives and even outperforms supervised solutions in some scenarios.

Large deviations for generalized Polya urns with arbitrary urn function

We consider a generalized two-color Polya urn (black and white balls) first introduced by Hill et al. (1980), where the urn composition evolves as follows: let π:0,1→0,1, and denote by xn the fraction of black balls after step n, then at step n+1 a black ball is added with probability πxn and a white ball is added with probability 1−πxn. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology.In this work we discuss large deviations for a wide class of continuous urn functions π. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit ϕs=limn→∞1nlogPnxn=sn,s∈0,1,where nxn is the number of black balls at time n, and use it to give some insight on the shape of ϕs. Under suitable assumptions on π we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi–Pal Model [6], giving the exact implicit expression for ϕ in terms of the Cumulant Generating Function.

Unsupervised Tracking With the Doubly Stochastic Dirichlet Process Mixture Model

We present an unsupervised tracking algorithm for human and car trajectory detection, using what is called the temporal doubly stochastic Dirichlet process (TDSDP) mixture model. The TDSDP captures the global dependence and the variation of human crowds and cars in temporal domains without the Markov assumption, making it particularly suitable for long-term tracking. Moreover, TDSDP prior can estimate the number of trajectories automatically. We first define the TDSDP based on the Cox process and then explain how to construct a TDSDP mixture model from thinning multiple Dirichlet process mixtures (DPMs) with conjugate priors. Next, a Markov chain Monte Carlo sampling inference is presented. Experimental results on synthetic and real-world data demonstrate that the proposed TDSDP mixture is superior to the DPM and the dependent Dirichlet process (DDP) in terms of topic variation modeling. PETS2001 data set experiments show that TDSDP has more robust object tracking capability over DDP based on generalized Polya urn. Low-quality fish data set experiments indicate that the TDSDP excels at solving tracking problems with insufficient features.

Technology and costs in international competitiveness: From countries and sectors to firms

This paper examines the microfoundations of the determinants of international competitiveness. It does so within the broader “technology gap” perspective whereby wide technological and organizational differences ultimately shape the patterns of trade within sectors across countries and their dynamics. First, we take stock of the incumbent evidence on the relation between cost-related and technological competition at country and sectoral level. The overall picture indeed suggests that the countries’ sectoral market shares are mainly shaped by technological factors while cost advantages/disadvantages do not seem to play any significant role. But within any sector, within any country, firms widely differ. Hence the question: does this property apply also at a micro level? Here, we first propose a heuristic model based on a generalized Polya urn process yielding such a property and, then, empirically attempt to identify the underlying dynamics at the firm level using a large panel of Italian firms, over nearly two decades. Results show that also at micro level in most sectors investments and patents correlate positively both with the probability of being an exporter and with the capacity to acquire and to increase exports, whereas labour costs show a negative effect only in some sectors. The result is reinforced when separating the short- and long-run effects, highlighting the predominant impact of technological proxies and basically the irrelevance of wage costs.

Convergence of generalized urn models to non-equilibrium attractors

Generalized Polya urn models have been used to model the establishment dynamics of a small founding population consisting of k different genotypes or strategies. As population sizes get large, these population processes are well-approximated by a mean limit ordinary differential equation whose state space is the k simplex. We prove that if this mean limit ODE has an attractor at which the temporal averages of the population growth rate is positive, then there is a positive probability of the population not going extinct (i.e. growing without bound) and its distribution converging to the attractor. Conversely, when the temporal averages of the population growth rate are negative along this attractor, the population distribution does not converge to the attractor. For the stochastic analog of the replicator equations which can exhibit non-equilibrium dynamics, we show that verifying the conditions for convergence and non-convergence reduces to a simple algebraic problem. We also apply these results to selection–mutation dynamics to illustrate convergence to periodic solutions of these population genetics models with positive probability.

A Modification of a Percentile Estimation Procedure Based on Generalized Polya Urns

Adaptive designs find an important application in the estimation of unknown percentiles for an underlying dose-response curve. A nonparametric adaptive design was suggested by Mugno et al. (2004) to simultaneously estimate multiple percentiles of an unknown dose-response curve via generalized Polya urns. In this article, we examine the properties of the design proposed by Mugno et al. (2004) when delays in observing responses are encountered. Using simulations, we evaluate a modification of the design under varying group sizes. Our results demonstrate unbiased estimation with minimal loss in efficiency when compared to the original compound urn design.

Asymptotics in Random Recursive Circuits*

A random recursive circuit is a combinatorial structure with a stochastic growth rule. These models are profusely used in computer science as data structures and in other fields, such as biostatistics, to model the spread of epidemics. The evolution of this random structure can be described by means of a generalized Polya urn model where different colors of balls represent different kind of nodes of the circuit. We prove that the process that represents the proportion of balls of each color fits a Robbins–Monro stochastic recurrence scheme and, then, sufficient conditions are established in order to obtain almost sure convergence and a central limit theorem for this process.

LISA: Locally Interacting Sequential Adsorption

We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This class covers, in particular, generalized Polya urn scheme, Dubins–Freedman random measures, and cooperative sequential adsorption models studied previously. Specifically, we address models where the distribution of a newly added particle is determined by the distance to the closest particle from the chosen one. We address boundedness of the processes and convergence properties of the corresponding sample measure. We show that, in general, the limiting measure is random when it exists and that this is the case for a wide class of almost surely bounded processes.

Coupon collector’s problems with statistical applications to rankings

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Polya urn sampling. In particular, usual Polya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with \(m\) kinds of coupons, the observed order of \(m\) kinds of coupons corresponds to a permutation of \(m\) letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of \(m\) letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the \(m\) kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.

A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits

We introduce a generalized Polya urn model with the feature that the evolution of the urn is governed by a function which may change depending on the stage of the process, and we obtain a Strong Law of Large Numbers and a Central Limit Theorem for this model, using stochastic recurrence techniques. This model is used to represent the evolution of a family of acyclic directed graphs, called random circuits, which can be seen as logic circuits. The model provides asymptotic results for the number of outputs, that is, terminal nodes, of this family of random circuits.

A New Randomized Pólya Urn Model

In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Polya urn, that is reinforced every time. A single urn contains a white balls, b black balls and evolves as follows: at discrete times n=1,2,…, we sample Mn balls and note their colors, say Rn are white and Mn- Rn are black. We return the drawn balls in the urn. Moreover, NnRn new white balls and Nn (Mn- Rn) new black balls are added in the urn. The numbers Mn and Nn are random variables. We show that the proportions of white balls forms a bounded martingale sequence which converges almost surely. Necessary and sufficient conditions for the limit to concentrate on the set {0,1} are given.

Trail formation in ants. A generalized Polya urn process

Faced with a choice of paths, an ant chooses a path with a higher concentration of pheromone. Subsequently, it drops pheromone on the path chosen. The reinforcement of the pheromone-following behavior favors the selection of an initially discovered path as the preferred path. This may cause a long path to emerge as the preferred path, were it discovered earlier than a shorter path. However, the shortness of the shorter path offsets some of the pheromone accumulated on the initially discovered longer path. In this paper, we model the trail formation behavior as a generalized Polya urn process. For k equal length paths, we give the distribution of pheromone at any time and highlight its sole dependence on the initial pheromone concentrations on paths. Additionally, we propose a method to incorporate the lengths of paths in the urn process and derive how the pheromone distribution alters on its inclusion. Analytically, we show that it is possible, under certain conditions, to reverse the initial bias that may be present in favor of paths that were discovered prior to the discovery of more efficient (shorter) paths. This addresses the Plasticity-Stability dilemma for ants, by laying out the conditions under which the system will remain stable or become plastic and change the path. Finally, we validate our analysis and results using simulations.

Modeling industrial evolution in geographical space

In this article we study a class of evolutionary models of industrial agglomeration with local positive feedbacks, which allow for a wide set of empirically testable implications. Their roots rest in the Generalized Polya Urn framework. Here, however, we build on a birth-death process over a finite number of locations and a finite population of firms. The process of selection among production sites that are heterogeneous in their ‘intrinsic attractiveness’ occurs under a regime of dynamic increasing returns depending on the number of firms already present in each location. The general model is presented together with a few examples of small economies which help to illustrate the properties of the model and characterize its asymptotic behavior. Finally, we discuss a number of empirical applications of our theoretical framework. The basic model, once taken to the data, is able to empirically disentangle the relative strength of technologically specific agglomeration drivers (affecting differently firms belonging to different industrial sectors in each location) from site-specific geographical forces (horizontally acting upon all sectors in each location).

Technology choice and timing with positive network effects

When two competing and incompatible products coexist in a market, potential users face a choice between the two products and the alternative of deferring the decision. This paper examines the choice between the two substitutes, where each one is subject to a positive network effect. That is, a user of one of the products experiences an increase in the value for each additional person using the same product. We examine this buy or wait problem, either for an individual or a manager making the investment on behalf of a firm, by formulating and analyzing a decision-theoretic model. To model the stochastic evolution of market share, we build on the generalized Polya urn of Arthur et al. [Arthur, W.B., Ermoliev, Y.M., Kaniovski, Y.M., 1987. Path-dependent processes and the emergence of macro-structure. European Journal Of Operational Research 30 (3), 294–303], allowing for composition of the market to affect not just the relative market shares but also the absolute growth rate of the market. We show that the optimal strategy is defined by a pair of market penetration thresholds that depend on the market composition. Looking at the effect of the network effects on the optimal strategy, we find that more pronounced network effects can either raise or lower the penetration thresholds.

Limit theorems for triangular urn schemes

We study a generalized Polya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depending on the ratio of the diagonal elements in the replacement matrix; the limit laws include normal, stable and Mittag-Leffler distributions as well as some less familiar ones. The results are in some cases similar to, but in other cases strikingly different from, the results for irreducible replacement matrices.

Generalized URN models of evolutionary processes

Generalized Polya urn models can describe the dynamics of finite populations of interacting genotypes. Three basic questions these models can address are: Under what conditions does a population exhibit growth? On the event of growth, at what rate does the population increase? What is the long-term behavior of the distribution of genotypes? To address these questions, we associate a mean limit ordinary differential equation (ODE) with the urn model. Previously, it has been shown that on the event of population growth, the limiting distribution of genotypes is a connected internally chain recurrent set for the mean limit ODE. To determine when growth and convergence occurs with positive probability, we prove two results. First, if the mean limit ODE has an “attainable” attractor at which growth is expected, then growth and convergence toward this attractor occurs with positive probability. Second, the population distribution almost surely does not converge to sets where growth is not expected and almost surely does not converge to “nondegenerate” unstable equilibria or periodic orbits of the mean limit ODE. Applications to stochastic analogs of the replicator equations and fertility-selection equations of population genetics are given.

Urn Models, Replicator Processes, and Random Genetic Drift

To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and balls of the same colors (strategies) are added or removed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive. On the event of nonextinction, three results are proven: (i) the number of balls increases asymptotically at a linear rate, (ii) the distribution x(n) of strategies at the nth update is a noisy Cauchy-Euler approximation to the mean limit ODE of the process, and (iii) the limit set of x(n) is almost surely a connected internally chain recurrent set for the mean limit ODE. Under a stronger set of assumptions, it is shown that for any attractor of the mean limit ODE there is a positive probability that the limit set for x(n) lies in this attractor. Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to overcome natural selection in large populations with the potential for rapid growth.

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.