  It is known that the Langevin dynamics used in MCMC is the gradient flow of
the KL divergence on the Wasserstein space, which helps convergence analysis
and inspires recent particle-based variational inference methods (ParVIs). But
no more MCMC dynamics is understood in this way. In this work, by developing
novel concepts, we propose a theoretical framework that recognizes a general
MCMC dynamics as the fiber-gradient Hamiltonian flow on the Wasserstein space
of a fiber-Riemannian Poisson manifold. The "conservation + convergence"
structure of the flow gives a clear picture on the behavior of general MCMC
dynamics. The framework also enables ParVI simulation of MCMC dynamics, which
enriches the ParVI family with more efficient dynamics, and also adapts ParVI
advantages to MCMCs. We develop two ParVI methods for a particular MCMC
dynamics and demonstrate the benefits in experiments.
