Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models

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. 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 . 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.