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

David Clancy, Hanbaek Lyu, Sebastien Roch, Allan Sly

TL;DR

This work analyzes the robustness of likelihood-based root-state reconstruction on CFN trees when edge flip probabilities are imperfectly known. It introduces a magnetization-based posterior estimator and proves that, deep inside the KS reconstructibility regime, the magnetization computed under a nearby parameter ${\hat{\boldsymbol{\theta}}}$ remains strongly aligned with the true root spin $\sigma_\rho$, with precise tail bounds distinguishing good, moderate, and severe reconstruction outcomes. The main contribution is a detailed, recursion-driven analysis on a three-level subtree that yields sharp control of error propagation through the magnetization recursion and conditional independence, providing theoretical justification for using MLE in ancestral reconstruction without exact branch-length knowledge. A practical byproduct is an approximate expression for the gradient of the population log-likelihood with respect to edge parameters, supporting coordinate ascent approaches for branch-length estimation on known phylogenies. These results inform phylogenetic inference by clarifying when likelihood-based ancestral reconstruction remains reliable under parameter misspecification and by guiding optimization strategies for branch-length estimation.

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.

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

TL;DR

This work analyzes the robustness of likelihood-based root-state reconstruction on CFN trees when edge flip probabilities are imperfectly known. It introduces a magnetization-based posterior estimator and proves that, deep inside the KS reconstructibility regime, the magnetization computed under a nearby parameter remains strongly aligned with the true root spin , with precise tail bounds distinguishing good, moderate, and severe reconstruction outcomes. The main contribution is a detailed, recursion-driven analysis on a three-level subtree that yields sharp control of error propagation through the magnetization recursion and conditional independence, providing theoretical justification for using MLE in ancestral reconstruction without exact branch-length knowledge. A practical byproduct is an approximate expression for the gradient of the population log-likelihood with respect to edge parameters, supporting coordinate ascent approaches for branch-length estimation on known phylogenies. These results inform phylogenetic inference by clarifying when likelihood-based ancestral reconstruction remains reliable under parameter misspecification and by guiding optimization strategies for branch-length estimation.

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.
Paper Structure (13 sections, 4 theorems, 79 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview and organization
  3. Statement of main results
  4. Assumptions
  5. Posterior-based ancestral state reconstruction: insensitivity to parameters
  6. Application: the gradient of the population log-likelihood on a known phylogeny
  7. Proof of main result (Theorem \ref{['thm:Robust']})
  8. Conditional independence of magnetizations
  9. Bounds on $q$
  10. Effect of recursion
  11. Base case: three-level subtree
  12. Proof of Theorem \ref{['thm:Robust']}
  13. Other proofs

Key Result

Theorem 2.3

There exist constants $\delta_{eqn:antiReconstruction}, c_{eqn:Reconstruct},C_{eqn:Reconstruct}, c_{eqn:antiReconstruction}, C_{eqn:antiReconstruction}>0$ depending only on the constants in assumption1 such that the following holds for any unrooted binary tree $T$ and $\delta\le \delta_{eqn:antiReco

Figures (2)

  • Figure 1: Sample spin configuration on the leaves of a rooted binary tree $T$ with $n = 100$ leaves. The root is in green, blue leaves have a state $+1$ and red leaves have a state $-1$. The parameters $\theta^*_e, \hat{\theta}_e \overset{i.i.d.}{\sim}\textup{Unif}([0.9,0.95])$ and the root magnetization $Z_u \approx 0.997$.
  • Figure 2: Histogram the 100,000 samples of unsigned magnetization $\sigma_uZ_u$ at the root of a descendant subtree $T_u$ with $n = 1000$ leaves. The horizontal axis is the unsigned magnetization, and the vertical axis is the normalized frequency plotted on a logarithmic scale. The parameters $\theta^*_e, \hat{\theta}_e \overset{i.i.d.}{\sim}\textup{Unif}([0.9,0.95])$. The two spikes around $\sigma_u Z_u \approx \pm 0.1$ correspond roughly to a flip on the edge from $u$ to one of its two children. The spike close to $\sigma_uZ_u\approx -1$ corresponds roughly to a flip on both edges from $u$ to its children. The red region represents severe failures of reconstruction, the blue region represents moderate failures of reconstruction, and the green region correspond to (successful) reconstruction.

Theorems & Definitions (22)

  • Definition 2.1: Restricted parameter spaces
  • Definition 2.2: Magnetization
  • Theorem 2.3: Insensitivity of magnetization to parameters
  • Remark 2.4
  • Lemma 2.5: Likelihood and magnetization
  • Theorem 2.6: Population log-likelihood landscape: gradient
  • Corollary 2.7: Initialization by coordinate maximization
  • Claim 3.1: Independence of unsigned magnetizations
  • proof
  • Claim 3.2: Magnetization: two strong agreeing inputs
  • ...and 12 more