##

The standard HMM

Following the notation of Rabiner [57],
there are observation times. At each time
,
there is a discrete state variable which takes one of
values
.
According to the Markovian assumption, the probability
distribution of depends only on the value
of . This is described compactly as a state transition
probability matrix whose elements represent the
probability that equals given
that equals . The initial state probabilities
are denoted , the probability that
equals .
It is a **hidden** Markov model because
the states are hidden from view; we cannot observe them.
But, we can observe
the random data which is generated
according to a PDF dependent on the state at time .
We denote the PDF of under state as
.

The complete set of model parameters that define the HMM are

The Baum-Welch algorithm calculates
new estimates given an observation sequence
and a previous estimate of .
The algorithm is composed of two parts: the forward/backward procedure,
and the reestimation of parameters.

**Subsections**
Baggenstoss
2017-05-19