  Rate processes are often modeled using Markov-State Models (MSM). Suppose you
know a prior MSM, and then learn that your prediction of some particular
observable rate is wrong. What is the best way to correct the whole MSM? For
example, molecular dynamics simulations of protein folding may sample many
microstates, possibly giving correct pathways through them, while also giving
the wrong overall folding rate, when compared to experiment. Here, we describe
Caliber Corrected Markov Modeling (C2M2): an approach based on the principle of
maximum entropy for updating a Markov model by imposing state- and trajectory-
based constraints. We show that such corrections are equivalent to asserting
position-dependent diffusion coefficients in continuous-time continuous-space
Markov processes modeled by a Smoluchowski equation. We derive the functional
form of the diffusion coefficient explicitly in terms of the trajectory-based
constraints. We illustrate with examples of 2D particle diffusion and an
overdamped harmonic oscillator.
