Analyzing Generalized P\'olya Urn Models using Martingales, with an Application to Viral Evolution

TL;DR

This paper develops a martingale-based approach to analyze generalized Pólya Urn models, deriving variance expressions, approximating probability distributions, and applying the method to estimate viral mutation rates from SARS-CoV-2 genomic data.

Contribution

It provides an exact variance formula for the RPW model, corrects previous results, and introduces a new method for estimating viral mutation rates from genomic data.

Findings

Derived an exact variance expression for RPW model.

Corrected earlier variance results in literature.

Applied the model to SARS-CoV-2 data for mutation rate estimation.

Abstract

The randomized play-the-winner (RPW) model is a generalized P\'olya Urn process with broad applications ranging from clinical trials to molecular evolution. We derive an exact expression for the variance of the RPW model by transforming the P\'olya Urn process into a martingale, correcting an earlier result of Matthews and Rosenberger (1997). We then use this result to approximate the full probability mass function of the RPW model for certain parameter values relevant to genetic applications. Finally, we fit our model to genomic sequencing data of SARS-CoV-2, demonstrating a novel method of estimating the viral mutation rate that delivers comparable results to existing scientific literature.