Expectation maximization and complex duration distributions for continuous time Bayesian networks

Continuous time Bayesian networks (CTBNs) describe structured stochastic processes with finitely many states that evolve over continuous time. A CTBN is a directed (possibly cyclic) dependency graph over a set of variables, each of which represents a finite state continuous time Markov process whose transition model is a function of its parents. We address the problem of learning the parameters and structure of a CTBN from partially observed data. We show how to apply expectation maximization (EM) and structural expectation maximization (SEM) to CTBNs. The availability of the EM algorithm allows us to extend the representation of CTBNs to allow a much richer class of transition durations distributions, known as phase distributions. This class is a highly expressive semi-parametric representation, which can approximate any duration distribution arbitrarily closely. This extension to the CTBN framework addresses one of the main limitations of both CTBNs and DBNs — the restriction to exponentially / geometrically distributed duration. We present experimental results on a real data set of people’s life spans, showing that our algorithm learns reasonable models — structure and parameters — from partially observed data, and, with the use of phase distributions, achieves better performance than DBNs.

Learning continuous time Bayesian networks

Continuous time Bayesian networks (CTBN) describe structured stochastic processes with finitely many states that evolve over continuous time. A CTBN is a directed (possibly cyclic) dependency graph over a set of variables, each of which represents a finite state continuous time Markov process whose transition model is a function of its parents. We address the problem of learning parameters and structure of a CTBN from fully observed data. We define a conjugate prior for CTBNs and show how it can be used both for Bayesian parameter estimation and as the basis of a Bayesian score for structure learning. Because acyclicity is not a constraint in CTBNs, we can show that the structure learning problem is significantly easier, both in theory and in practice, than structure learning for dynamic Bayesian networks (DBNs). Furthermore, as CTBNs can tailor the parameters and dependency structure to the different time granularities of the evolution of different variables, they can provide a better fit to continuous-time processes than DBNs with a fixed time granularity.

Expectation Propagation for Continuous . . .

Continuous time Bayesian networks (CTBNs) describe structured stochastic processes with finitely many states that evolve over continuous time. A CTBN is a directed (possibly cyclic) dependency graph over a set of variables, each of which represents a finite state continuous time Markov process whose transition model is a function of its parents. As shown previously, exact inference in CTBNs is intractable. We address the problem of approximate inference, allowing for general queries conditioned on evidence over continuous time intervals and at discrete time points. We show how CTBNs can be parameterized within the exponential family, and use that insight to develop a message passing scheme in cluster graphs and allows us to apply expectation propagation to CTBNs. The clusters in our cluster graph do not contain distributions over the cluster variables at individual time points, but distributions over trajectories of the variables throughout a duration. Thus, unlike discrete time temporal models such as dynamic Bayesian networks, we can adapt the time granularity at which we reason for different variables and in different conditions