### 6.5.2 Hidden Markov Models

A **hidden Markov model (HMM)** is an augmentation of the Markov
chain to include observations. Just like the state transition of the
Markov chain, an HMM also includes
observations of the state. These observations can be **partial** in that
different states can map to the same observation and **noisy** in that the
same state can stochastically map to different observations at different times.

The assumptions behind an HMM are that the state at time *t+1* only depends on the state at time *t*, as in
the Markov chain. The observation at
time *t* only depends on the state at time *t*. The observations are
modeled using the variable *O _{t}* for each time

*t*whose domain is the set of possible observations. The belief network representation of an HMM is depicted in Figure 6.14. Although the belief network is shown for four stages, it can proceed indefinitely.

A stationary HMM includes the following probability distributions:

*P(S*specifies initial conditions._{0})*P(S*specifies the dynamics._{t+1}|S_{t})*P(O*specifies the sensor model._{t}|S_{t})

There are a number of tasks that are common for HMMs.

The problem of **filtering** or belief-state **monitoring** is to determine the current state
based on the current and previous observations, namely to determine

P(S_{i}|O_{0},...,O_{i}).

Note that all state and observation variables after
*S _{i}* are irrelevant because they are not observed and can be ignored when this
conditional distribution is computed.

The problem of **smoothing** is to determine a state
based on past and future observations. Suppose an agent has observed up to
time *k* and wants to determine the state at time *i* for
*i<k*; the smoothing problem is to determine

P(S_{i}|O_{0},...,O_{k}).

All of the variables *S _{i}* and

*V*for

_{i}*i>k*can be ignored.