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

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.