Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models

TL;DR

This paper studies high-dimensional mean estimation in a binary Markov Gaussian mixture model, deriving near-optimal error bounds and proposing an algorithm for joint estimation of the mean and Markov chain parameter.

Contribution

It introduces a new high-dimensional estimation framework that accounts for Markovian memory effects and provides nearly minimax optimal error bounds with an algorithm for unknown parameters.

Findings

Derived nearly minimax optimal error rates for mean estimation.

Provided tight bounds for estimating the Markov chain flip probability.

Developed an algorithm with theoretical guarantees for joint estimation of mean and Markov parameter.

Abstract

We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an estimator observes $n$ samples of a $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$, multiplied by a random sign $ S_i $ ($1\le i\le n$), and corrupted by isotropic standard Gaussian noise. The sequence of signs $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with flip probability $\delta\in[0,1/2]$. As $\delta$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $\delta=0$ and the Gaussian Mixture Model for which $\delta=1/2$. Assuming that the estimator knows $\delta$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|\theta_{*}\|,\delta,d,n$. We then provide an upper bound to the case of estimating $\delta$, assuming a (possibly inaccurate) knowledge of $\theta_{*}$. The bound is proved to be tight when $\theta_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $\theta_{*}$ with $\delta$ unknown a priori, and theoretical guarantees on its error are stated.