Markov state models revisited: Principles and algorithms for unbiased observables

What the math gives to ML

The paper identifies a concrete source of coarse-graining bias: a single transition matrix cannot generally represent both equilibrium observables and source-sink nonequilibrium observables at a fixed lag and coarse partition. Its transferable mechanism is to maintain two weighted transition operators, one estimated under equilibrium sampling and one under recycling dynamics, and to match each observable to the operator that generated its ensemble. In neural networks, this suggests a dual-kernel latent world model or recurrent state-space model whose equilibrium head predicts stationary statistics while a source-sink head predicts first-passage, hitting, and directed-flow quantities without requiring excessively long lags.