Likelihood landscape of binary latent model on a tree

TL;DR

The paper analyzes edge-length estimation in the Cavender-Farris-Neyman (CFN) model on trees, recasting the problem as maximizing a non-convex likelihood with hidden states. It develops a population-level second-order analysis around the true edge-parameter vector ${\boldsymbol{\theta}}^{*}$, showing that the expected log-likelihood is $\Theta(\delta^{-1})$-strongly concave and $\Theta(\delta^{-1})$-smooth in an $L_ox$ of radius $\Theta(\delta)$, with a box size independent of tree topology or leaf count. Central to the argument is a detailed Hessian decomposition in terms of magnetizations, where diagonal entries dominate and off-diagonal terms decay exponentially with graph distance, enabling a Gershgorin-based conclusion of global strong concavity and a unique maximizer at ${\boldsymbol{\theta}}^{*}$. The work introduces a robust magnetization-insensitivity principle, together with a decomposition of the sample space into signal tiers, and four key technical lemmas that bound complex products of dependent terms. Together, these results provide the first rigorous justification for the effectiveness of gradient- or coordinate-based optimization methods in latent-tree likelihood problems and offer a blueprint for broader latent-variable analyses on tree-structured graphs.

Abstract

We study the optimization landscape of maximum likelihood estimation for a binary latent tree model with hidden variables at internal nodes and observed variables at the leaves. This model, known as the Cavender-Farris-Neyman (CFN) model in statistical phylogenetics, is also a special case of the ferromagnetic Ising model. While the likelihood function is known to be non-concave with multiple critical points in general, gradient descent-type optimization methods have proven surprisingly effective in practice. We provide theoretical insights into this phenomenon by analyzing the population likelihood landscape in a neighborhood of the true parameter vector. Under some conditions on the edge parameters, we show that the expected log-likelihood is strongly concave and smooth in a box around the true parameter whose size is independent of both the tree topology and number of leaves. The key technical contribution is a careful analysis of the expected Hessian, showing that its diagonal entries are large while its off-diagonal entries decay exponentially in the graph distance between the corresponding edges. These results provide the first rigorous theoretical evidence for the effectiveness of optimization methods in this setting and may suggest broader principles for understanding optimization in latent variable models on trees.

Figures (5)

• Figure 1: Steel's example. There are four leaves, the top two vertices have states $\sigma_v^{(1)} = +1$, $\sigma^{(2)}_v = -1$, and the bottom two vertices have $\sigma^{(1)}_u = -1$, $\sigma^{(2)}_u = +1$.
• Figure 2: Decomposition of the tree $T$ into subtrees with respect to (a) a single edge $e=\{x,y\}$ and (b) two edges $e=\{y_{N}, y_{N+1}\}$ and $f=\{ y_{-1}, y_{0} \}$. For instance, the subtree $T_{u}$ in panel a is rooted at $u$ and $L_{u}$ denotes the set of all leaves in $T_{u}$; The subtree $T_{y_{N}}$ in panel b is rooted at $y_{N+1}$ and contains all nodes $y_{-1},\dots,y_{N}$ and $w_{0},\dots,w_{N}$ as well as the subtrees $T_{y_{-1}},T_{w_{0}},\dots,T_{W_{N}}$.
• Figure 3: Propagation of signals from the edge $f=\{y_{0},y_{-1}\}$ to $e=\{y_{N},y_{N+1}\}$ when $N=7$. The signal $\xi_{j+1}$ that $y_{j+1}$ receives from $y_{j}$ is the scalar multiple $\hat{\theta}_{j}$ of the magnetization $Z_{y_{j}}=q(\xi_{j},\eta_{j})$ at $y_{j}$.
• Figure 4: Signals used to define $W_{4}$ in \ref{['eq:def_W_i']}. Note that $W_{4}$ does not depend on $\xi_{1}$ due to the adversarialization.
• Figure 5: Depiction of the random variables $\xi_{0},\xi_{1},\eta_{0},\eta_{1},\eta_{2}$ associated with the $N=1$ case.

Theorems & Definitions (42)

• Definition 2.1: Restricted parameter spaces
• Theorem 2.2: Population log-likelihood landscape: Hessian
• Corollary 2.3: Population log-likelihood landscape: strong concavity and smoothness
• Definition 3.1: Magnetization
• Proposition 3.2: Likelihood and magnetization
• proof
• Theorem 3.3: Insensitivity of magnetization to parameters
• Claim 3.4: Independence of unsigned magnetization
• Corollary 3.5
• proof