Hamiltonian Monte Carlo from $q$-deformed phase-space mechanics

Abstract Hamiltonian Monte Carlo (HMC) constructs Markov transitions by combiningHamiltonian dynamics with a Metropolis correction. This paper develops a\(q\)-deformed HMC framework motivated by \(q\)-deformed phase-space mechanics.Starting from a Lagrangian formulation, we derive the corresponding\(q\)-Hamiltonian equations and show the formal invariance of the associated\(q\)-symplectic form. This provides a geometric origin for the proposed\(q\)-HMC dynamics rather than treating it as an ad hoc finite-differencemodification.To obtain a computable sampler for ordinary target distributions, we realizethe formal \(q\)-differential operators through Jackson-type differences andconstruct a Metropolis-adjusted \(q\)-Hamiltonian proposal. The resultingtransition reduces to classical HMC as \(q\to1\), satisfies detailed balanceunder the Metropolis correction, and admits sufficient nondegeneracy conditionsfor irreducibility of the marginal position chain.Numerical experiments on benchmark targets, positive-scale black-box targets,a mixed positive/signed target, and a PDE-based Bayesian inverse problem showthat the \(q\)-Jackson force acts as a relative perturbation mechanism in theoriginal variables. It avoids the domain violations and scale mismatch thatcan occur with raw additive finite differences, while tracking exact-gradient,adjoint-force, or carefully scaled finite-difference benchmarks when these areavailable.