For the sake of mathematical and computational tractability, following assumptions are made in the theory of HMMs.
In other words it is assumed that
the next state is dependent only upon the current state. This is
called the Markov assumption and the resulting model becomes
actually a first order HMM.
However generally the next state may depend on past k states and it is possible to obtain a such model, called an order HMM by defining the transition probabilities as follows.
But it is seen that a higher order HMM will have a higher complexity. Even though the first order HMMs are the most common, some attempts have been made to use the higher order HMMs too.
for any and .
. Then according to the assumption for an HMM ,
However unlike the other two, this assumption has a very limited validity. In some cases this assumption may not be fair enough and therefore becomes a severe weakness of the HMMs.