Sample Complexity of Branch-length Estimation by Maximum Likelihood

2025-07-29

AI Summary

This paper addresses maximum-likelihood estimation of branch lengths on binary trees under leaf-based repeated observations. To overcome the challenge that marginal transition probabilities are unobservable in latent-variable tree models, we propose a coordinate-ascent algorithm and—crucially—establish, for the first time, that within the Kesten–Stigum reconstructibility regime, the log-likelihood function exhibits a strongly concave region with high probability, ensuring geometric convergence of the algorithm. Our approach integrates Markov tree models, stochastic analysis, and sample-complexity analysis under balanced tree topologies. Theoretically, with polynomial sample size (m), the algorithm converges to a neighborhood of the true parameters with high probability, achieving estimation error (O(1/sqrt{m})). Empirical experiments corroborate both practical efficacy and theoretical consistency. The core contribution lies in rigorously linking tree reconstructibility, the geometric landscape of the likelihood function, and the convergence guarantees of optimization algorithms.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider the branch-length estimation problem on a bifurcating tree: a character evolves along the edges of a binary tree according to a two-state symmetric Markov process, and we seek to recover the edge transition probabilities from repeated observations at the leaves. This problem arises in phylogenetics, and is related to latent tree graphical model inference. In general, the log-likelihood function is non-concave and may admit many critical points. Nevertheless, simple coordinate maximization has been known to perform well in practice, defying the complexity of the likelihood landscape. In this work, we provide the first theoretical guarantee as to why this might be the case. We show that deep inside the Kesten-Stigum reconstruction regime, provided with polynomially many $m$ samples (assuming the tree is balanced), there exists a universal parameter regime (independent of the size of the tree) where the log-likelihood function is strongly concave and smooth with high probability. On this high-probability likelihood landscape event, we show that the standard coordinate maximization algorithm converges exponentially fast to the maximum likelihood estimator, which is within $O(1/sqrt{m})$ from the true parameter, provided a sufficiently close initial point.

Problem

• Estimating branch-lengths in binary trees using maximum likelihood
• Analyzing non-concave log-likelihood landscapes in phylogenetics
• Proving convergence of coordinate maximization to true parameters

Innovation

• Uses two-state symmetric Markov process
• Applies coordinate maximization algorithm
• Ensures strong concavity in likelihood landscape