Efficient inference in dynamic belief networks with variable temporal resolution

Dynamic Belief Networks (DBNs) have been used for the monitoring and control of stochastic dynamical processes where it is crucial to provide a response in real-time. DBN transition functions are typically specified as conditional probability distributions over a constant time interval. When these functions are used to model dynamic systems with observations that occur at irregular intervals, both exact and approximate DBN inference algorithms are inefficient. This is because the computation of the posterior distribution at an arbitrary time in the future involves repeated application of the fixed time transition model. We draw on research from mathematics and theoretical physics that shows the dynamics inherent to a Markov model can be described as a diffusion process. These systems can be modelled using the Fokker-Planck equation, the solutions of which are the transition functions of the system for arbitrary length time intervals. We show that using these transition functions in a DBN inference algorithm gives significant computational savings compared to the traditional constant time-step model.