Learning parameters of hybrid time Bayesian networks

Time granularity is an important factor in characterizing dynamical systems. Hybrid time Bayesian networks model the dynamics of systems that contain both irregularly-timed variables and variables whose evolution is naturally described by discrete time. The former observations are modeled as variables in continuous-time manner and the latter are modeled by discrete-time random variables. We address the problem of learning parameters of hybrid time models from complete data where all the states are known at any time point, and from incomplete trajectories, where continuous-time variables are observed only at some time points. We show that for the complete case, the parameters can be estimated straightforwardly. When some continuous-time variables are (partially) unobserved, it becomes infeasible to learn the parameters in closed form. In that case, we propose to use Markov chain Monte Carlo sampling to estimate the posterior distribution over the parameters. We tested the approach on a number of hybrid time models where continuous-time variables are completely or partially observed, showing that close estimation of the original parameters can be recovered. A medical example is used to illustrate the learning parameters of hybrid time Bayesian networks.