The fundamental assumption
of an HMM is that the process to be modeled is governed by a finite
number of states and that these states change
once per time step in a random but statistically
predictable way. To be more precise, let
be the probability that the system
transitions into state at time .
The Markovian assumption says that
depends only on , the true state at time .
Furthermore, if this distribution does not depend
on the absolute time , then the state probabilities can be
described completely by a fixed state transition matrix
Figure 13.20 illustrates
a hidden Markov model (HMM). At each time step (time running from left to right),
the Markov model is in one of the five possible
states. According to the Markovian assumption, the probability
that the model is in state at time is governed
only by the transition probability , where is the true
state at time . The Markov model is
``hidden" from view by the observer who can only
observe measurements whose PDF is governed by
the true state at each time step.
The mathematics of the HMM are reviewed in section 13.3.2.
A hidden Markov model (HMM). As
the state transitions occur from sample to sample,
the observer, cannot see the states directly.
Instead, the observer makes observations whose PDF depends
on the state.