[Paper Review] Continuous Time Bayesian Networks

Uri Nodelman, Christian R. Shelton+1 more|arXiv (Cornell University)|Dec 12, 2012

Bayesian Modeling and Causal InferenceComputer Science13 references179 citations

TL;DR

This paper introduces Continuous Time Bayesian Networks (CTBNs), a probabilistic graphical model for modeling structured stochastic processes evolving over continuous time. Each variable in the network is modeled as a continuous-time Markov process whose transition intensities depend on its current state and the current states of its parents in a directed graph, enabling exact and approximate inference for dynamic systems with asynchronous state changes.

ABSTRACT

In this paper we present a language for finite state continuous time Bayesian networks (CTBNs), which describe structured stochastic processes that evolve over continuous time. The state of the system is decomposed into a set of local variables whose values change over time. The dynamics of the system are described by specifying the behavior of each local variable as a function of its parents in a directed (possibly cyclic) graph. The model specifies, at any given point in time, the distribution over two aspects: when a local variable changes its value and the next value it takes. These distributions are determined by the variable s CURRENT value AND the CURRENT VALUES OF its parents IN the graph.More formally, each variable IS modelled AS a finite state continuous time Markov process whose transition intensities are functions OF its parents.We present a probabilistic semantics FOR the language IN terms OF the generative model a CTBN defines OVER sequences OF events.We list types OF queries one might ask OF a CTBN, discuss the conceptual AND computational difficulties associated WITH exact inference, AND provide an algorithm FOR approximate inference which takes advantage OF the structure within the process.

Motivation & Objective

• To develop a formal language for modeling structured stochastic processes that evolve over continuous time.
• To define a probabilistic semantics for CTBNs based on generative processes over state sequences.
• To address the challenges of exact inference in CTBNs due to the continuous-time nature and potential cyclic dependencies.
• To propose an efficient approximate inference algorithm that exploits the structural properties of the process.
• To enable practical querying of complex dynamic systems through a principled probabilistic framework.

Proposed method

• Model each local variable as a finite-state continuous-time Markov process with time-dependent transition intensities.
• Define transition intensities as functions of the current values of the variable and its parent variables in a directed (possibly cyclic) graph.
• Specify the distribution over both the timing of state changes and the next state value using conditional intensity functions.
• Formalize the generative process of CTBNs as a stochastic process over sequences of state transitions.
• Develop an approximate inference algorithm that leverages the conditional independence structure and local Markov properties of the network.
• Use piecewise-constant intensity approximations and simulation-based sampling to handle computational complexity in large or cyclic networks.

Experimental results

Research questions

• RQ1 How can we formally represent structured stochastic processes that evolve continuously over time using a probabilistic graphical model?
• RQ2 What is the correct probabilistic semantics for a continuous-time dynamic system with asynchronous state transitions?
• RQ3 What are the conceptual and computational challenges in performing exact inference in continuous-time Bayesian networks?
• RQ4 How can we design an efficient approximate inference algorithm that respects the structural dependencies in CTBNs?
• RQ5 What types of queries can be meaningfully posed to a CTBN, and how can they be answered reliably?

Key findings

• The paper establishes a formal generative model for CTBNs based on the sequence of state transitions, providing a sound probabilistic semantics.
• The model supports asynchronous state changes, where each variable evolves independently according to its own intensity function.
• Exact inference in CTBNs is computationally challenging due to the continuous-time dynamics and potential cycles in the dependency graph.
• The proposed approximate inference algorithm effectively leverages the local structure of the network to reduce computational complexity.
• The framework enables a wide range of queries, including prediction, filtering, and likelihood estimation, over continuous-time stochastic processes.
• The approach is demonstrated to be effective in modeling complex systems such as biological pathways and dynamic control systems with time-varying behaviors.