I have a system that can be simulated with a Markov process $M_1$, but it could also be simulated with a higher order Markov process $M_n$, which has memory for 1 or more previous states. Is there a way to measure how well $M_1$ versus $M_n$ represents/captures information in the system?
I would like to generate simulated data from a data set I have, and I think a Markov (high order or not) process is likely the best way to do it, but I want an explicit measure for how well different types of Markov processes replicate the data I am interested in. Perhaps there may be something in information theory that can measure this.