Robust estimation of latent tree graphical models: Inferring hidden states with inexact parameters

Latent tree graphical models are widely used in computational biology, signal and image processing, and network tomography. Here we design a new efficient, estimation procedure for latent tree models, including Gaussian and discrete, reversible models, that significantly improves on previous sample requirement bounds. Our techniques are based on a new hidden state estimator which is robust to inaccuracies in estimated parameters. More precisely, we prove that latent tree models can be estimated with high probability in the so-called Kesten-Stigum regime with $O(log^2 n)$ samples where $n$ is the number of nodes.

💡 Research Summary

This paper addresses the fundamental problem of learning latent tree graphical models—both discrete reversible Markov random fields (GTR models) and Gaussian Markov random fields on trees (GMRFT)—under the Kesten‑Stigum (KS) regime, where the maximal edge weight τ⁺ is below the critical threshold g★KS = ln√2. Two intertwined tasks are considered: Tree Model Estimation (TME), i.e., recovering the tree topology and edge lengths from leaf observations, and Hidden State Inference (HSI), i.e., estimating the hidden root state given a fully specified model and a single leaf sample.

Prior work achieved optimal O(log n) sample complexity for TME only under the restrictive assumption that edge lengths are discretized, a device used to avoid the sensitivity of HSI to parameter errors. The authors remove this assumption by designing a new hidden‑state estimator that remains robust when the transition parameters are only approximately known.

The algorithm follows a “boosting” framework. An initial quartet‑based step reconstructs the first level of the tree using only leaf data. Then, iteratively for each subsequent level, (i) hidden states at the roots of already reconstructed sub‑trees are inferred, and (ii) a one‑level TME step uses these inferred states to grow the tree upward. The core technical contribution is a recursive estimator for internal nodes. For a node v with children y₁ and y₂, the estimator takes the form S_v = ω₁ S_{y₁} + ω₂ S_{y₂}, where the weights ω₁, ω₂ are chosen based on previously estimated edge lengths and measured bias terms B(y₁), B(y₂). The construction guarantees (a) conditional unbiasedness up to a bias factor B(v) that stays close to 1, and (b) an exponential‑moment bound E