Assumptions and definitions.

The basic assumption in the Markov models is that the system has a finite number of possible states, but it can occupy only one of them at a time. In each state characteristic signals are emitted and the signal features are observed independently for time intervals $t=1,\ldots,T$.For a sequence of observed feature vectors of the system, $\mbox{\boldmath$O$}=(\mbox{\boldmath$O$}_1,\mbox{\boldmath$O$}_2,\ldots,\mbox{\boldmath$O$}_T)\:$,there exists then the corresponding hidden sequence of states, $q = (q_0,q_1,\ldots,q_T)$,that the system has visited starting from the initial state q₀. Another basic assumption is the Markov property, i.e. the transition probability of the system to the next state depends only on the previous state regardless of the earlier transition history.

The HMM consists of the state transition probability matrix $A=[a_{ij}] \:$, where

$$a_{ij} = \Pr(q_t = j\vert q_{t-1} = i) \:,\; i,j = 1, \ldots, N \:,$$ (13)

and the set of state probability density models $B=\{b_i(\mbox{\boldmath$O$}_t)\}_{i=1}^N \:$.The density model can be either a set of discrete probabilities for quantized observations  

$$b_i(\mbox{\boldmath$O$}_t) = \Pr(\mbox{\boldmath$O$}_t\vert q_t = i)$$ (14)

or a continuous probability density function  

$$b_i(\mbox{\boldmath$O$}_t) = P(\mbox{\boldmath$O$}_t\vert q_t = i) \:.$$ (15)

Combining the definitions above and the starting probabilities of the states $\pi_i = \Pr(q_0 = i)$, an HMM can then be described fully by the triple   [Rabiner, 1989,Juang and Rabiner, 1991]. Different HMMs can be conveniently connected by including the transitions between models to the matrix A. For example, in ASR one HMM is usually assigned to each phoneme, but it is also possible to use indeclinable words or other parts of words as the basic units.