Existing rigorous convergence guarantees for the Hamiltonian Monte Carlo (HMC) algorithm use Gaussian auxiliary momentum variables, which are crucially symmetrically distributed. We present a novel convergence analysis for HMC utilizing new analytic and probabilistic arguments. The convergence is rigorously established under significantly weaker conditions, which among others allow for general auxiliary distributions. In our framework, we show that plain HMC with asymmetrical momentum distributions breaks a key self-adjointness requirement. We propose a modified version that we call the Alternating Direction HMC (AD-HMC). Sufficient conditions are established under which AD-HMC exhibits geometric convergence in Wasserstein distance. Numerical experiments suggest that AD-HMC can show improved performance over HMC with Gaussian auxiliaries.
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