  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.
