Conditional-path Monte Carlo for rare stochastic dynamics on networks: Details and derivations

What the math gives to ML

The paper provides a constructive path-space Monte Carlo mechanism for rare stochastic trajectories: represent a full trajectory as a graph of local events and perform non-local Swendsen-Wang-like cluster updates that preserve path constraints while avoiding catastrophic rejection. Its transferable assets are uniformization-based detailed-balance moves, tunable nonnegative event rates that maximize trajectory mobility, and dynamic programming for exact boundary-conditioned path generation. The strongest neural-network application is constrained sampling and training of long-horizon neural state-space or world models, where ordinary autoregressive sampling almost never produces a specified rare terminal event. These methods make quantitative predictions about valid-sample rates, autocorrelation times, and constraint violations.