foundations of computational agents
The third edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2023 is now available (including full text).
One of the problems with the definition of an HMM or a dynamic belief network is that the model depends on the time granularity. The time granularity specifies how often a dynamic system transitions from one state to the next. The time granularity could either be fixed, for example each day or each thirtieth of a second, or it could be event based, where a time step occurs when something interesting occurs. If the time granularity were to change, for example from daily to hourly, the conditional probabilities would also change.
One way to model the dynamics independently of the time granularity is to model continuous time, where for each variable and each value for the variable, the following are specified:
a distribution of how long the variable is expected to keep that value (e.g., an exponential decay) and
what value it will transition to when its value changes.
Given a discretization of time, where time moves from one state the next in discrete steps, a dynamic belief network can be constructed from this information. If the discretization of time is fine enough, ignoring multiple value transitions in each time step will result only in small errors.