Conditional Expectation and the Approximation of Labelled Markov Processes

We develop a new notion of approximation of labelled Markov processes based on the use of conditional expectations. The key idea is to approximate a system by a coarse-graining of the state space and using averages of the transition probabilities. This is unlike any of the previous notions based on the use of simulation. The resulting approximations are customizable, more accurate and stay within the world of LMPs. The use of averages and expectations may well also make the approximations more robust. We introduce a novel condition called “granularity” which leads to unique conditional expectations and which turns out to be a key concept despite its simplicity.