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Traditionally, the field of computational Bayesian statistics has beendivided into two main subfields: variational methods and Markov chain MonteCarlo (MCMC). In recent years, however, several methods have been proposedbased on combining variational Bayesian inference and MCMC simulation in orderto improve their overall accuracy and computational efficiency. This marriageof fast evaluation and flexible approximation provides a promising means ofdesigning scalable Bayesian inference methods. In this paper, we explore thepossibility of incorporating variational approximation into a state-of-the-artMCMC method, Hamiltonian Monte Carlo (HMC), to reduce the required gradientcomputation in the simulation of Hamiltonian flow, which is the bottleneck formany applications of HMC in big data problems. To this end, we use a {\itfree-form} approximation induced by a fast and flexible surrogate functionbased on single-hidden layer feedforward neural networks. The surrogateprovides sufficiently accurate approximation while allowing for fastexploration of parameter space, resulting in an efficient approximate inferencealgorithm. We demonstrate the advantages of our method on both synthetic andreal data problems.

Variational Hamiltonian Monte Carlo via Score Matching