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 _*R^d, multiplied by a random sign S_i (1 i n), and corrupted by isotropic standard Gaussian noise. The sequence of signs _i\_i[n]\-1,1\^n is drawn from a stationary homogeneous Markov chain with flip probability [0,1/2]. As varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which =0 and the Gaussian Mixture Model for which =1/2. Assuming that the estimator knows , we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of \|_*\|,,d,n. We then provide an upper bound to the case of estimating , assuming a (possibly inaccurate) knowledge of _*. The bound is proved to be tight when _* is an accurately known constant. These results are then combined to an algorithm which estimates _* with unknown a priori, and theoretical guarantees on its error are stated.

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Reproductions