We consider a high-dimensional mean estimation problem for binary hidden Markov models that reveals the interplay between data memory, sample size, dimensionality, and signal strength in statistical inference. In this model, the estimator observes $n samples of the $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$ and randomly signs $ S_i $ ($1\ le i\le n$) and is corrupted by isotropic standard Gaussian noise. sequence of symbols $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with reversal probability $\delta\in[0,1/2]As $.$\delta$ varies, this model smoothly interpolates between two well-studied models: the Gaussian position model with $\delta=0$ and the Gaussian mixture with $\delta=1/2$. Model. Assuming that the estimator knows $\delta$, as a function of $\|\theta_{*}\|,\delta,d,n$, the approximate minimum optimal estimation error (up to a logarithmic coefficient) establish a rate. We then provide an upper bound on which to estimate $\delta$ given (possibly imprecise) knowledge of $\theta_{*}$. If $\theta_{*}$ is exactly a known constant, it proves that the bounds are narrow. These results are combined into an algorithm that estimates $\theta_{*}$ with $\delta$ unknown in advance, and a theoretical guarantee against that error is stated.
