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

TL;DR

This paper derives the posterior distribution of a random process conditioned on empirical frequencies, showing that for i.i.d. sequences and finite Markov chains, the distribution can be identified through empirical data, linking it to concepts like Gibbs conditioning and data-driven entropy.

Contribution

It provides a unified framework for understanding the posterior distribution of processes conditioned on empirical frequencies, extending to both i.i.d. and Markov cases with a novel connection to thermodynamic formalism.

Findings

Posterior distribution can be identified via empirical frequencies.

Conditional symmetry simplifies the posterior analysis.

Introduces a data-driven entropy concept using Large Deviations Principles.

Abstract

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.