The state at a particular time need not be represented as a single variable. It is often more natural to represent the state in terms of features.

A dynamic belief network (DBN) is a discrete time belief network with regular repeated structure. It is like a (hidden) Markov model, but the states and the observations are represented in terms of features. If $F$ is a feature, we write $F_t$ as the random variable that represented the value of variable $F$ at time $t$. A dynamic belief network makes the following assumptions:

• •

  The set of features is the same at each time.

• •

  For any time $t>0$, the parents of variable $F_t$ are variables at time $t$ or time $t-1$, such that the graph for any time is acyclic. The structure does not depend on the value of $t$ (except $t=0$ is a special case).

• •

  The conditional probability distribution of how each variable depends on its parents is the same for every time $t>0$. This is called a stationary model.

Thus, a dynamic belief network specifies a belief network for time $t=0$, and for each variable $F_t$ specifies $P(F_t\mid parents(F_t))$, where the parents of $F_t$ are in the same or previous time steps. This is specified for $t$ as a free parameter; the conditional probabilities can be used for any time $t>0$. As in a belief network, directed cycles are not allowed.

The model for a dynamic belief network is represented as a two-step belief network, which represents the variables at the first two times (times 0 and 1). That is, for each feature $F$ there are two variables, $F_0$ and $F_1$. The set of parents of $F_0$, namely $parents(F_0)$ can only include variables for time 0. The resulting graph must be acyclic. Associated with the network are the probabilities $P(F_0\mid parents(F_0))$ and $P(F_1\mid parents(F_1))$.

The two-step belief network is unfolded into a belief network by replicating the structure for subsequent times. In the unfolded network, $P(F_i\mid parents(F_i))$, for $i>1$, has exactly the same structure and the same conditional probability values as $P(F_1\mid parents(F_1))$.