[Paper Review] Expectation Propagation for Continuous Time Bayesian Networks

Uri Nodelman, Daphne Koller+1 more|arXiv (Cornell University)|Jul 4, 2012

Bayesian Modeling and Causal Inference9 references67 citations

TL;DR

This paper proposes an approximate inference method for Continuous Time Bayesian Networks (CTBNs) using Expectation Propagation (EP) on cluster graphs with trajectory-based clusters. By parameterizing CTBNs within the exponential family and adapting time granularity per variable, the method enables efficient, scalable inference over continuous time intervals and discrete evidence points, overcoming the intractability of exact inference in CTBNs.

ABSTRACT

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.

Motivation & Objective

• To address the intractability of exact inference in Continuous Time Bayesian Networks (CTBNs), which model structured stochastic processes over continuous time.
• To develop a scalable and flexible approximate inference framework that supports general queries over continuous time intervals and discrete time points.
• To enable variable-specific time granularity in reasoning by modeling distributions over entire variable trajectories rather than individual time points.
• To apply Expectation Propagation (EP) to CTBNs by constructing a cluster graph with trajectory-based clusters and leveraging exponential family parameterization.

Proposed method

• The authors parameterize CTBNs within the exponential family to enable efficient message passing using sufficient statistics.
• They design a cluster graph where each cluster represents a distribution over the trajectories of variables over a continuous time interval, rather than at discrete time points.
• Message passing is performed across clusters using Expectation Propagation to approximate the true posterior distribution over trajectories.
• The method allows for adaptive time granularity per variable, dynamically adjusting resolution based on the dynamics and evidence.
• The algorithm supports general queries conditioned on evidence at discrete time points and over continuous time intervals.
• The approach leverages the structure of CTBNs and the properties of exponential families to maintain computational tractability while improving approximation accuracy.

Experimental results

Research questions

• RQ1 How can approximate inference in CTBNs be made scalable and efficient while supporting continuous time evidence?
• RQ2 Can trajectory-based clusters in a cluster graph improve the accuracy and flexibility of inference in CTBNs compared to discrete-time models?
• RQ3 To what extent can time granularity be adapted per variable in CTBN inference without sacrificing computational efficiency?
• RQ4 How does the proposed EP-based method compare to existing approximate inference techniques in terms of convergence and accuracy on CTBN queries?
• RQ5 Can the exponential family parameterization of CTBNs be effectively exploited to enable message passing with sufficient statistics in a cluster-based framework?

Key findings

• The proposed EP-based inference method achieves accurate approximation of posterior distributions in CTBNs over both continuous time intervals and discrete time evidence points.
• By modeling distributions over entire trajectories, the method enables variable-specific time granularity, improving efficiency and precision where needed.
• The use of exponential family parameterization allows for efficient message passing using sufficient statistics, reducing computational overhead.
• The cluster graph structure supports scalable inference by localizing computation to relevant time intervals and variable dependencies.
• The method demonstrates practical applicability to complex, real-world stochastic processes where exact inference is infeasible.
• Empirical results show that the approach converges reliably and provides competitive accuracy compared to baseline methods on benchmark CTBN queries.