Markov Property

States do not depend on evidence.

$X_t$ only depends on $X_{0:t-1}$ and not on the evidence variables $E_{1:t}$.

Other special cases:

First-order Markov Property: $\mathrm{P}\left({\mathbf{X}}_t\mid {\mathbf{X}}_{0:t-1}\right)=\mathrm{P}\left({\mathbf{X}}_t\mid {\mathbf{X}}_{t-1}\right)$ The current Random Variable only depends on the previous random variable. The Markov Chain is an example of a first-order Markov Process. It has no memory.

Second-order Markov Property: $\mathrm{P}\left({\mathbf{X}}_t\mid {\mathbf{X}}_{0:t-1}\right)=\mathrm{P}\left({\mathbf{X}}_t\mid {\mathbf{X}}_{t-2},{\mathbf{X}}_{t-1}\right)$ The current Random Variable only depends on the previous two random variables.

Increasing the order adds memory to the process.