Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models

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 $θ_{*}\in\mathbb{R}^{d}$, 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 $δ\in[0,1/2]$. As $δ$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $δ=0$ and the Gaussian Mixture Model for which $δ=1/2$. Assuming that the estimator knows $δ$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|θ_{*}\|,δ,d,n$. We then provide an upper bound to the case of estimating $δ$, assuming a (possibly inaccurate) knowledge of $θ_{*}$. The bound is proved to be tight when $θ_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $θ_{*}$ with $δ$ unknown a priori, and theoretical guarantees on its error are stated.