What does Markov assumption state about the history of sequences?

A stochastic process has the Markov property if the probability distribution of future states conditioned on both the present and past states depends only on the present state or, more formally, the following equality holds.

  p(st+1∣st,st−1:1)=p(st+1∣st),∀t

The hidden Markov model (HMM) is an example of a model where the Markov property is often assumed to hold. In other words, the Markov assumption is made in the case of the HMM.There are also the variable-order (or higher-order) Markov models, where the future state can depend on the previously n states or, more formally, the following equality holds.

 p(st+1∣st,st−1:1)=p(st+1∣st:t−n),∀t In this context, a hidden Markov model is called a first-order Markov model (n=0). Therefore, there can also be second-order (n=1), third-order (n=2), etc., Markov models. In fact, there are also higher-order hidden Markov models. To conclude, the expressions Markov property and Markov assumption are not exactly interchangeable. The Markov property is an attribute that a stochastic process can be assumed to possess. In that case, the Markov assumption is made. The expression Markov property usually refers to a first-order Markov property, but it can more generally refer to a higher-order Markov property.