Orbital MCMC
This work addresses a fundamental limitation in MCMC algorithm design for probabilistic inference, offering incremental improvements to enhance efficiency and flexibility.
The paper tackles the problem of designing deterministic transformations for MCMC proposals under invariance constraints, deriving a new acceptance test for arbitrary maps and proposing two practical algorithms based on periodic orbits and contractions. The empirical study demonstrates practical advantages of these kernels.
Markov Chain Monte Carlo (MCMC) algorithms ubiquitously employ complex deterministic transformations to generate proposal points that are then filtered by the Metropolis-Hastings-Green (MHG) test. However, the condition of the target measure invariance puts restrictions on the design of these transformations. In this paper, we first derive the acceptance test for the stochastic Markov kernel considering arbitrary deterministic maps as proposal generators. When applied to the transformations with orbits of period two (involutions), the test reduces to the MHG test. Based on the derived test we propose two practical algorithms: one operates by constructing periodic orbits from any diffeomorphism, another on contractions of the state space (such as optimization trajectories). Finally, we perform an empirical study demonstrating the practical advantages of both kernels.