HMMs are generally assumed to have a discrete-valued hidden state,
with a multinomial distribution for
(given the previous values of the state).
In this paper, we sometimes use the slight abuse of
language often found in papers on HMMs for speech recognition
and talk about different states instead of different values
of the state random variable.
In the discrete state case, if the model is homogeneous,
the transition parameters (for Markov
models of order 1) can be represented by a
matrix of transition probabilities
[4]
.
In section 7 we discuss continuous
state models, also called state-space models [&make_named_href('',
"node31.html#kitagawa+gersch96","[47]")],
in which the next-state probability distribution is usually
a Gaussian whose mean is a linear function of the
previous state. We also briefly discuss hybrids with both
discrete and continuous state variables.
In section 5 we present models in which the transitions are not only conditioned on the previous state but also on an ``input variable'', i.e., the sequence of transition probabilities is obtained as a time-independent function of a time-varying input sequence.