Nonparametric Bayesian analysis for the Galton-Watson process

TL;DR

This paper introduces a nonparametric Bayesian method using Dirichlet Process priors for the Galton-Watson process, enabling flexible inference on offspring distributions and improved classification, demonstrated through simulations and COVID-19 data analysis.

Contribution

It develops a fully non-parametric Bayesian approach for the Galton-Watson process, allowing learning of the offspring distribution support and handling overdispersion.

Findings

The DP prior improves classification accuracy with complete and incomplete data.

Simulation results show the method's robustness and flexibility.

Application to COVID-19 data demonstrates practical utility.

Abstract

The Galton-Watson process is a model for population growth which assumes that individuals reproduce independently according to the same offspring distribution. Inference usually focuses on the offspring average as it allows to classify the process with respect to extinction. We propose a fully non-parametric approach for Bayesian inference on the GW model using a Dirichlet Process prior. The prior naturally generalizes the Dirichlet conjugate prior distribution, and it allows learning the support of the offspring distribution from the data as well as taking into account possible overdispersion of the data. The performance of the proposed approach is compared with both frequentist and Bayesian procedures via simulation. In particular, we show that the use of a DP prior yields good classification performance with both complete and incomplete data. A real-world data example concerning COVID-19 data from Sardinia illustrates the use of the approach in practice.