Markov processes follow from the principle of Maximum Caliber

TL;DR

This paper demonstrates that Markov processes naturally emerge from the principle of Maximum Caliber, establishing a theoretical foundation and showing how MaxCal can be used for parameter inference in stochastic dynamics.

Contribution

It proves that Markov models are the unique maximizers of path entropy under various data conditions, linking MaxCal to maximum likelihood estimation.

Findings

Markov processes maximize path entropy given certain data constraints.

MaxCal provides a method to compute model parameters equivalent to maximum likelihood.

Markov models arise as the natural outcome of the principle of Maximum Caliber.

Abstract

Markov models are widely used to describe processes of stochastic dynamics. Here, we show that Markov models are a natural consequence of the dynamical principle of Maximum Caliber. First, we show that when there are different possible dynamical trajectories in a time-homogeneous process, then the only type of process that maximizes the path entropy, for any given singlet statistics, is a sequence of identical, independently distributed (i.i.d.) random variables, which is the simplest Markov process. If the data is in the form of sequentially pairwise statistics, then maximizing the caliber dictates that the process is Markovian with a uniform initial distribution. Furthermore, if an initial non-uniform dynamical distribution is known, or multiple trajectories are conditioned on an initial state, then the Markov process is still the only one that maximizes the caliber. Second, given a model, MaxCal can be used to compute the parameters of that model. We show that this procedure is equivalent to the maximum-likelihood method of inference in the theory of statistics.