Likelihood-Based Root State Reconstruction on a Tree: Sensitivity to Parameters and Applications

TL;DR

This paper analyzes the sensitivity of maximum likelihood root state reconstruction on trees under the CFN model, showing robustness when true flip probabilities are small and providing insights for phylogenetic branch length estimation.

Contribution

It demonstrates that MLE remains accurate under parameter uncertainty when flip probabilities are low, supporting its use in phylogenetics and offers a new approximation for likelihood gradients.

Findings

MLE agrees with true root state when flip probabilities are small

Posterior root magnetization is robust to parameter estimation errors

Provides a new approximation for likelihood gradient in leaf state models

Abstract

We consider a broadcasting problem on a tree where a binary digit (e.g., a spin or a nucleotide's purine/pyrimidine type) is propagated from the root to the leaves through symmetric noisy channels on the edges that randomly flip the state with edge-dependent probabilities. The goal of the reconstruction problem is to infer the root state given the observations at the leaves only. Specifically, we study the sensitivity of maximum likelihood estimation (MLE) to uncertainty in the edge parameters under this model, which is also known as the Cavender-Farris-Neyman (CFN) model. Our main result shows that when the true flip probabilities are sufficiently small, the posterior root mean (or magnetization of the root) under estimated parameters (within a constant factor) agrees with the root spin with high probability and deviates significantly from it with negligible probability. This provides theoretical justification for the practical use of MLE in ancestral sequence reconstruction in phylogenetics, where branch lengths (i.e., the edge parameters) must be estimated. As a separate application, we derive an approximation for the gradient of the population log-likelihood of the leaf states under the CFN model, with implications for branch length estimation via coordinate maximization.