arXiv subjects

David Clancy Jr.

Publications and source records attributed to David Clancy Jr..

3 recordsLinked to original sources

arXiv2025

Sample Complexity of Branch-length Estimation by Maximum Likelihood

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.

stat.CO↗

arXiv2025

Likelihood landscape of binary latent model on a tree

We investigate the optimization landscape of maximum likelihood estimation (MLE) for the Cavender-Farris-Neyman (CFN) model, a two-state latent tree model fundamental to statistical phylogenetics and the ferromagnetic Ising model. Although the log-likelihood function is non-concave and may admit many critical points, simple coordinate maximization algorithms are remarkably effective in practice. We provide the first theoretical justification for this success. We prove that sufficiently deep inside the reconstruction regime, the population log-likelihood is strongly concave and smooth within a box around the true parameter, whose size is independent of tree topology and number of leaves. This fundamental result implies that the empirical landscape shares these regularity properties with high probability given polynomial sample complexity and also that coordinate maximization converges exponentially fast to an $O(1/\sqrt{m})$-consistent MLE. Our analysis centers on a novel decay property of the population Hessian: diagonal entries remain large while off-diagonal entries decay exponentially with graph distance. These results provide rigorous theoretical evidence for the efficacy of likelihood-based tree inference and suggest broader principles for latent variable models.

math.ST↗

arXiv2025

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

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.

math.PR↗