2022/02/23 by Wenqing Hu, Hong Qian, Hu, Wenqing +1
Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2202.11780
openalex publication_date 2022/02/23 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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.