Nonparametric Bayesian analysis for the Galton-Watson process

2025-06-26

AI Summary

This paper addresses two key challenges in Galton–Watson branching processes: unknown offspring distribution forms and frequent overdispersion in observed data. To tackle these, we propose the first fully nonparametric Bayesian inference framework, employing a Dirichlet process prior on the offspring distribution to jointly and adaptively learn its support, shape, and overdispersion structure. Posterior inference for extinction probability is implemented via MCMC, and the method maintains high classification accuracy even under incomplete observations. Simulation studies and analysis of real-world Sardinian COVID-19 incidence data demonstrate that our approach substantially outperforms classical frequentist and parametric Bayesian methods—particularly in sparse and truncated data settings—exhibiting strong robustness and inferential reliability. By avoiding restrictive parametric assumptions, the framework establishes a scalable, nonparametric paradigm for branching process modeling, enabling flexible and data-driven inference in complex epidemic and population dynamics applications.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Nonparametric Bayesian inference for Galton-Watson offspring distribution
• Addressing overdispersion and support learning in population growth models
• Comparing DP prior performance with frequentist and Bayesian methods

Innovation

• Nonparametric Bayesian approach for Galton-Watson process
• Dirichlet Process prior for offspring distribution
• Handles overdispersion and incomplete data effectively