Why are DBNs sparse?

Real stochastic processes operating in continuous time can be modeled by sets of stochastic differential equations. On the other hand, several popular model families, including hidden Markov models and dynamic Bayesian networks (DBNs), use discrete time steps. This paper explores methods for converting DBNs with infinitesimal time steps into DBNs with finite time steps, to enable efficient simulation and filtering over long periods. An exact conversion—summing out all intervening time slices between two steps— results in a completely connected DBN, yet nearly all human-constructed DBNs are sparse. We show how this sparsity arises from well-founded approximations resulting from differences among the natural time scales of the variables in the DBN. We define an automated procedure for constructing an approximate DBN model for any desired time step and prove error bounds for the approximation. We illustrate the method by generating a series of approximations to a simple pH model for the human body, demonstrating speedups of several orders of magnitude compared to the original model.