论文标题
关于以有限路径的经验频率为条件的随机过程的后验分布：I.I.D和有限马尔可夫链案例
On the Posterior Distribution of a Random Process Conditioned on Empirical Frequencies of a Finite Path: the i.i.d and finite Markov chain case
论文作者
论文摘要
我们获得以观察有限样本路径的经验频率为条件的随机过程的后验分布。我们发现，在对过程的“依赖性结构”（{\ em c.f.}独立性或马尔可夫人的“依赖性结构”的相当广泛的假设下，可以将过程索引的后边缘分布确定为从样本路径的观察到的经验频率中计算出的某些经验分布。我们表明，在这两种离散价值I.I.D.的情况下序列和有限马尔可夫链，观察经验频率给出了某种“条件对称性”，这会导致后验分布的所需结果。有限时间观察及其渐近无限时间限制的结果都通过吉布斯调节的想法连接。最后，由于我们的结果证明了经验频率在理解数据的信息内容中的核心作用，因此我们使用大偏差原理（LDP）来构建“数据驱动熵”的一般概念，从中可以将统计热力学统计热力学的最新研究从中应用到数据。
We obtain the posterior distribution of a random process conditioned on observing the empirical frequencies of a finite sample path. We find under a rather broad assumption on the "dependence structure" of the process, {\em c.f.} independence or Markovian, the posterior marginal distribution of the process at a given time index can be identified as certain empirical distribution computed from the observed empirical frequencies of the sample path. We show that in both cases of discrete-valued i.i.d. sequence and finite Markov chain, a certain "conditional symmetry" given by the observation of the empirical frequencies leads to the desired result on the posterior distribution. Results for both finite-time observations and its asymptotic infinite-time limit are connected via the idea of Gibbs conditioning. Finally, since our results demonstrate a central role of the empirical frequency in understanding the information content of data, we use the Large Deviations Principle (LDP) to construct a general notion of "data-driven entropy", from which one can apply a formalism from the recent study of statistical thermodynamics to data.
