Asymptotics of posteriors for binary branching processes

TL;DR

This paper analyzes the asymptotic behavior of posterior distributions for initial populations and parameters in binary branching processes, comparing Bayesian and naive methods, and providing central limit theorems with explicit variances.

Contribution

It introduces a Bayesian approach to estimate initial populations and parameters in binary branching processes and compares it with a naive hitting time method, deriving asymptotic distributions.

Findings

Posterior distributions converge to normal distributions with explicit variances.

Bayesian and naive methods have comparable asymptotic properties.

Explicit central limit theorems are established for both procedures.

Abstract

We compute the posterior distributions of the initial population and parameter of binary branching processes, in the limit of a large number of generations. We compare this Bayesian procedure with a more na\"ive one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances.