Beyond Reweighting: Certified Improvements from New Transitions in Discrete MCMC

arXiv:2611.15668v1 Announce Type: new Abstract: Optimizing transition probabilities on a fixed grap may leave substantial limitations on the efficiency of reversible Markov chain Monte Carlo (MCMC). We develop a framework for transition-support design that considers both allowed transitions and their stationary flows. For a fixed target distribution, observable, and node-flow budget, we derive a necessary and sufficient dual condition for support augmentation to strictly reduce the optimal value of the asymptotic variance, together with computable primal--dual certificates that establish improvements over every admissible original-support reweighting. We also construct a family in which adding a single edge reduces integrated autocorrelation time by an $\Omega(n)$ factor relative to optimal original-support weighting, demonstrating that structural limitations can impose arbitrarily large relative losses in estimation efficiency. To realize these gains, we investigate energy-informed edge selection and constrained stationary-flow optimization. Experiments on molecular and Ising state graphs distinguish the potential value of new transitions from the performance of kernels constructed on them: on identical augmented Ising supports, full weight optimization certifies strict improvements in 24/24 conditions, whereas degree-based Metropolis--Hastings weighting certifies none. Our results provide a principled approach to characterizing, certifying, and exploiting transition-support improvements in explicitly represented finite state spaces.