慢混合长记忆马尔可夫过程的平稳和转移概率
Stationary and Transition Probabilities in Slow Mixing, Long Memory Markov Processes
摘要：
我们观察到一个长度为$n$的样本，它是由一个未知的平稳遍历马尔可夫过程（\emph{model}）在有限字母表$\mathcal{A}$上生成的.给定来自$\mathcal{A}$的符号的任何字符串$\bf{w}$，我们想要估计跟随$\bf{w}$的符号的条件概率分布，以及$\bf{w}$的平稳概率。在此设置中使估计复杂化的两个不同问题是（i）长存储器，以及（ii）甚至仅用一位存储器也可能发生的\emph{slow mixing}。在这种情况下，任何一致估计只能在所有遍历马尔可夫模型的类上逐点收敛。也就是说，给定任何估计量和任何样本大小$n$，基础模型可能使得估计量在具有高概率的大小为$n$的样本上表现不佳。但是我们能不能看一个长度$n$样本，并确定\emph{if}的估计可能是准确的？由于内存是未知的\emph{a-priori}，因此自然的方法是估计具有内存$k_n=\mathcal{O}(\log n)$的可能更粗糙的模型。随着$n$的增长，只要$k_n$的标度在$n$中不是超对数的，那么最终几乎肯定成立的逐点一致估计（eas）是已知的。在这里，我们不希望得到简单的收敛结果，而是希望得到长度为$n$的样本的最佳答案。将泛压缩的结果与Aldous的耦合参数相结合，我们得到了长度- $n$样本（即使是慢混合模型）的充分条件，以确定何时朴素（i）条件概率的估计和（ii）与平稳概率相关的估计是准确的;并且还限制了朴素估计与真实值的偏差。
英文摘要：
We observe a length-$n$ sample generated by an unknown,stationary ergodic Markov process (\emph{model}) over a finite alphabet $\mathcal{A}$. Given any string $\bf{w}$ of symbols from $\mathcal{A}$ we want estimates of the conditional probability distribution of symbols following $\bf{w}$, as well as the stationary probability of $\bf{w}$. Two distinct problems that complicate estimation in this setting are (i) long memory, and (ii) \emph{slow mixing} which could happen even with only one bit of memory. Any consistent estimator in this setting can only converge pointwise over the class of all ergodic Markov models. Namely, given any estimator and any sample size $n$, the underlying model could be such that the estimator performs poorly on a sample of size $n$ with high probability. But can we look at a length-$n$ sample and identify \emph{if} an estimate is likely to be accurate? Since the memory is unknown \emph{a-priori}, a natural approach is to estimate a potentially coarser model with memory $k_n=\mathcal{O}(\log n)$. As $n$ grows, pointwise consistent estimates that hold eventually almost surely (eas) are known so long as the scaling of $k_n$ is not superlogarithmic in $n$. Here, rather than eas convergence results, we want the best answers possible with a length-$n$ sample. Combining results in universal compression with Aldous' coupling arguments, we obtain sufficient conditions on the length-$n$ sample (even for slow mixing models) to identify when naive (i) estimates of the conditional probabilities and (ii) estimates related to the stationary probabilities are accurate; and also bound the deviations of the naive estimates from true values.
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