Dimension Reduction in Singularly Perturbed Continuous-Time Bayesian Networks

Continuous-time Bayesian networks (CTBNs) are graphical representations of multi-component continuous-time Markov processes as directed graphs. The edges in the network represent direct influences among components. The joint rate matrix of the multi-component process is specified by means of conditional rate matrices for each component separately. This paper addresses the situation where some of the components evolve on a time scale that is much shorter compared to the time scale of the other components. In this paper, we prove that in the limit where the separation of scales is infinite, the Markov process converges (in distribution, or weakly) to a reduced, or effective Markov process that only involves the slow components. We also demonstrate that for reasonable separation of scale (an order of magnitude) the reduced process is a good approximation of the marginal process over the slow components. We provide a simple procedure for building a reduced CTBN for this effective process, with conditional rate matrices that can be directly calculated from the original CTBN, and discuss the implications for approximate reasoning in large systems.

💡 Research Summary

Continuous‑time Bayesian networks (CTBNs) provide a compact graphical representation of multi‑component continuous‑time Markov processes. Each node corresponds to a variable, and its dynamics are described by a conditional rate matrix that depends on the current states of its parent nodes. The global rate matrix of the joint process is obtained by a tensor‑product‑like composition of these local matrices, which makes CTBNs especially suitable for modeling complex systems with many interacting components.

In many realistic domains, however, the components evolve on widely different time scales. Some variables change orders of magnitude faster than others (e.g., fast biochemical reactions versus slow gene‑expression dynamics, or high‑frequency clock signals versus slower data‑transfer events). When such a multi‑scale structure is ignored, the full CTBN becomes computationally burdensome: the global rate matrix is stiff, simulation steps must be extremely small, and inference algorithms suffer from severe numerical instability.

The paper tackles this problem by formalising the notion of a singularly perturbed CTBN. The authors partition the set of variables into a “fast” subset (Y) and a “slow” subset (X). The overall generator (L_{\varepsilon}) of the joint process is written as

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