The Rescaled Pólya Urn and the Wright—Fisher Process with Mutation

Giacomo Aletti,Irene Crimaldi

doi:10.3390/math9222909

Giacomo Aletti, Irene Crimaldi

Open Access

https://doi.org/10.3390/math9222909

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Abstract

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright—Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

Highlights

The well-known standard Eggenberger—Pólya urn [1,2] works as follows
We prove that the multidimensional Wright— Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated with a family of rescaled Pólya (RP) urns with β ∈[0,1), β → 1
The known properties of the WF process can give a description of the RP urn when the parameter β is strictly smaller than one, but very near to one

Summary

Introduction

The well-known standard Eggenberger—Pólya urn [1,2] works as follows. An urn initially contains N0, i balls of color i, for i = 1, . . . , k, and at each time-step, a ball is drawn from the urn and it is returned into the urn together with α > 0 additional balls of the same color (here and in the following, the expression “number of balls” is not to be understood literally, but all the quantities are real numbers, not necessarily integers). At time-step 0, the urn contains bi + B0, i > 0 balls of color i and the parameters α > 0 and β ≥ 0 regulate the reinforcement mechanism. The WF diffusion processes are widely employed in Bayesian statistics, as models for time-evolving priors [8,9,10,11] and as a discrete-time finite-population construction method of the two-parameter Poisson– Dirichlet diffusion [12] They have been applied in genetics [13,14,15,16,17,18], in biophysics [19,20], in filtering theory [21,22] and in finance [23,24].

The Rescaled Pólya Urn

Main Result

Recessive and Dominant Colors in an RP Urn with β Near to 1

Conclusions