Markov Property

States do not depend on evidence.

XtX_t only depends on X0:t−1X_{0:t-1} and not on the evidence variables E1:tE_{1:t}.

Other special cases:

First-order Markov Property:

P(Xt∣X0:t−1)=P(Xt∣Xt−1)\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:

P(Xt∣X0:t−1)=P(Xt∣Xt−2,Xt−1)\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.