Representing Markov processes as dynamic non-parametric Bayesian networks

We prove that a k-th order Markov process has a dynamic NPBN representation. Guidance is given on how to obtain the various dependence metrics that are sufficient and necessary. We additionally derive the conditions required to perform conditioning which can be analytically done for the Gaussian case. One of the advantages consists in having a clear vision on the dependence dynamics expressed through the time copula and rank correlation. Compared to classic stochastic process based modelling, this may shed light on non-stationarity concerning dependence. It thus enhances the description/characterization of dependencies. More precisely, for Levy processes whose increments are independent and stationary, the associated time-copula may thus be non-stationary as is shown taking the example of the Brownian motion. The applicability of the Markov process representation may find interest in various fields ranging from finance, where Markov processes such as the geometric Brownian motion is key for stock pricing, to deterioration modelling, speech recognition, etc. Basically, these are the areas into which Markovian features have been successfully tested and validated. In this regard, we illustrate our findings through an example focused around the Brownian motion