Ancestral Inference and Learning for Branching Processes in Random Environments

TL;DR

This work develops a generalized method of moments framework for ancestral inference in branching processes in random environments (BPRE), enabling estimation of the ancestral mean $m_A$ and the marginal offspring mean $m^*$ from descendant-generation data. The authors show that, under multiple sampling schemes, the estimators exhibit asymptotic normality with a scaling that involves the environment’s coefficient of variation, and that the ancestral and offspring estimators become independent in the limit. They provide variance estimation strategies under Beta-Binomial and Poisson-Gamma models, and validate the approach through extensive numerical experiments and real data from PCR quantitation and early COVID-19 spread in the US. A bootstrap procedure for confidence intervals is proposed to handle finite-sample regimes, and the methods are demonstrated to offer practical improvements over competing models such as LSTAR in PCR quantitation and epidemic-growth contexts. Overall, the paper delivers a rigorous statistical toolkit for ancestral inference in BPREs with broad applicability to biological assays and infectious-disease dynamics, including robust uncertainty quantification and data-driven learning strategies.

Abstract

Ancestral inference for branching processes in random environments involves determining the ancestor distribution parameters using the population sizes of descendant generations. In this paper, we introduce a new methodology for ancestral inference utilizing the generalized method of moments. We demonstrate that the estimator's behavior is critically influenced by the coefficient of variation of the environment sequence. Furthermore, despite the process's evolution being heavily dependent on the offspring means of various generations, we show that the joint limiting distribution of the ancestor and offspring estimators of the mean, under appropriate centering and scaling, decouple and converge to independent Gaussian random variables when the ratio of the number of generations to the logarithm of the number of replicates converges to zero. Additionally, we provide estimators for the limiting variance and illustrate our findings through numerical experiments and data from Polymerase Chain Reaction experiments and COVID-19 data.

Figures (13)

• Figure 1: Example of the PCR dataset
• Figure 2: VA Population
• Figure 3: Weekly Accumulated Patients for Counties in VA Grouped by Population
• Figure 4: case (i)
• Figure 5: case (ii)

Theorems & Definitions (49)

• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Remark 2.2: Comparison to GW case
• Theorem 3.1
• Theorem 3.2
• Remark 3.1
• Theorem 3.3