Input-Output Hidden Markov Models (IOHMMs) [&make_named_href('',
"node31.html#Bengio+Frasconi-nips7-iohmms","[5]")]
(or Conditional HMMs) are simply HMMs for which the emission and transition
distributions are conditional on another sequential random variable, called
the input sequence, and that we will note
. In that case, the
observations modeled with the emission distributions are called outputs, and
the model represents not the distribution of sequences
but instead
the conditional distribution
. In the simpler models first
presented here, the input and output sequences have the same length, but a
recent extension [&make_named_href('',
"node31.html#Bengio+Bengio96","[93]")] (section 5.3) allows
input and output sequences of different lengths. Transducers
(section 6) which can be seen as generalizations of such
conditional distributions, also allow input and output sequences of different
lengths. The conditional independence assumption of a synchronous IOHMM are
represented in the probabilistic graphical model of Figure 10.
Figure 10: Probabilistic graphical model representing
the independence assumptions of a synchronous
Input-Output Hidden Markov Model.
The state sequence is
, the output
sequence is
, and the input sequence
is
.
In the simpler case in which the input and output sequences are synchronous,
the mathematics of IOHMMs are very similar to that of HMMs but more general.
For this reason we will explain the EM algorithm used to train HMMs (and
IOHMMs) in this section (in subsection 5.2). Whereas in ordinary
HMMs the emission distributions are given by a homogeneous model
, in IOHMMs, they are given by a time-varying conditional model
(i.e., the emission density associated to each state
changes
in function of the input
). Similarly, instead of time-invariant
transition probabilities
, in IOHMMs we have
. More generally, values of the inputs
at different time steps around
can be
used to condition these distributions. The ``forward'' phase
copmutation in IOHMMs is identical in form to that in HMMs:
Whereas HMMs used for pattern
recognition are often trained by choosing parameters
to maximize the
likelihood of the acoustic observations given the correct classification
sequence,
, IOHMMs may be trained to maximize the
likelihood
of decision variables
given the
actually observed (e.g., acoustic) variables
.
In the literature on learning
algorithms [&make_named_href('',
"node31.html#Cacciatore-nips94","[4]"), &make_named_href('',
"node31.html#Bengio+Frasconi-nips7-iohmms","[5]"), &make_named_href('',
"node31.html#Meila96","[6]")],
IOHMMs have been proposed for sequence processing tasks, with
complex emission and transition models based on ANNs.
There are also earlier treatments of similar models in
statistics [&make_named_href('',
"node31.html#Lindgren78","[94]")] and econometrics [&make_named_href('',
"node31.html#Hamilton89","[12]")].
In the control and reinforcement learning literature, similar models
have been called Partially Observable Markov Decision
Processes [&make_named_href('',
"node31.html#Sondik73","[95]"), &make_named_href('',
"node31.html#Sondik78","[96]"), &make_named_href('',
"node31.html#Chrisman92AAAI","[3]")].
In this case, the objective is not to model an output sequence
given an input sequence, but rather, to find the input
sequence (in fact a sequence of actions)
which minimizes a cost function defined
on the sequence of outputs (which are observed).
In this case the IOHMM represents the probabilistic
relationship between the actions and the observations,
with a hidden state variable. Although the HMMs such
as those described in section 3 for
speech recognition do represent the conditional probability
of an observation sequence given
a classification sequence, this is achieved by
deterministically attaching a symbolic meaning to
the different values of the state variable. Instead,
in IOHMMs, the state variable is stochastically
related to the output variable, and for classification
problems, one can view the output sequence as the
classification sequence and the input (observed)
sequence as the conditioning sequence.
Potential advantages of IOHMMs over HMMs can be summarized as follows (and are described in more detail at the end of this section): the training criterion is discriminant, the emission and transition models can be naturally represented by flexible models such as ANNs, the problem of imbalance between transition and emission probabilities is reduced, and long-term dependencies may be more easily captured and learned.
In the next section we describe particular kinds of IOHMMs which have been proposed in the econometrics literature. In the following section, we present the EM algorithm which can be used for training both HMMs and synchronous HMMs.