大状态空间马氏链的熵率估计
Entropy Rate Estimation for Markov Chains with Large State Space
摘要：
基于数据的熵估计是分布性质检验和估计中的典型问题之一。为了估计具有独立样本的$S$元素上的分布的香农熵，[Paninski 2004]表明$S$中的样本复杂性是次线性的，[Valiant-Valiant 2011]表明当且仅当样本大小$n$远远超过$\frac{S}{\log S}$时，香农熵的一致估计是可能的。本文考虑了由$n$观测的样本路径估计具有$S$状态的平稳可逆马氏链的熵率的问题。我们证明：（1）只要马尔可夫链混合速度不太慢，即，弛豫时间至多为$O(\frac{S}{\ln^3 S})$，当$n \gg \frac{S^2}{\log S}$时可实现一致估计。(2)只要马尔可夫链有一些轻微的依赖性，即，弛豫时间至少为$1+Ω(\frac{\ln^2 S}{\sqrt{S}})$，当$n \lesssim \frac{S^2}{\log S}$时不可能进行一致的估计。在这两种假设下，最佳的估计精度是$Θ(\frac{S^2}{n \log S})$。相比之下，经验熵率要求至少$Ω(S^2)$样本是一致的，即使当马尔可夫链是无记忆的。除了合成实验之外，我们还应用实现最佳样本复杂度的估计器来估计Penn Treebank和Google One Billion Words语料库中英语语言的熵率，这为语言建模提供了一个自然的基准，并将其直接与广泛使用的困惑度度量相关联。
英文摘要：
Estimating the entropy based on data is one of the prototypical problems in distribution property testing and estimation. For estimating the Shannon entropy of a distribution on $S$ elements with independent samples, [Paninski2004] showed that the sample complexity is sublinear in $S$, and [Valiant--Valiant2011] showed that consistent estimation of Shannon entropy is possible if and only if the sample size $n$ far exceeds $\frac{S}{\log S}$. In this paper we consider the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations. We show that: (1) As long as the Markov chain mixes not too slowly, i.e., the relaxation time is at most $O(\frac{S}{\ln^3 S})$, consistent estimation is achievable when $n \gg \frac{S^2}{\log S}$. (2) As long as the Markov chain has some slight dependency, i.e., the relaxation time is at least $1+Ω(\frac{\ln^2 S}{\sqrt{S}})$, consistent estimation is impossible when $n \lesssim \frac{S^2}{\log S}$. Under both assumptions, the optimal estimation accuracy is shown to be $Θ(\frac{S^2}{n \log S})$. In comparison, the empirical entropy rate requires at least $Ω(S^2)$ samples to be consistent, even when the Markov chain is memoryless. In addition to synthetic experiments, we also apply the estimators that achieve the optimal sample complexity to estimate the entropy rate of the English language in the Penn Treebank and the Google One Billion Words corpora, which provides a natural benchmark for language modeling and relates it directly to the widely used perplexity measure.
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