论文标题
高维二进制马尔可夫高斯混合模型中的平均估计
Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models
论文作者
论文摘要
我们考虑了二进制隐藏的马尔可夫模型上的高维平均估计问题，该模型阐明了数据，样本大小，维度和统计推断中信号强度的记忆之间的相互作用。 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.标志的顺序$ \ {s_ {i} \} _ {i \ in [n]} \ in \ { - 1,1,1 \}^{n} $从具有flip概率$δ\ in [0,1/2] $中的固定同质马尔可夫链中绘制。随着$δ$的变化，该模型平稳地插入了两个认真的模型：高斯定位模型，$δ= 0 $和高斯混合模型，$δ= 1/2 $。假设估计器知道$δ$，我们建立了一个几乎最小的最佳（超过对数因素）估计错误率，这是$ \ |θ_ {*} \ |，δ，d，n $的函数。然后，我们为估计$δ$的情况提供了上限，假设有$θ_ {*} $的知识（可能不准确）。当$θ_ {*} $是一个准确已知的常数时，该界限被证明是紧密的。然后将这些结果组合到一个算法，该算法估计$θ_ {*} $，$δ$未知的先验性，并说明了其错误的理论保证。
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.
