Abstract
Sampling using integrator-dependent shadow Hamiltonian’s has been shown to produce improved sampling properties relative to Hamiltonian Monte Carlo. The shadow Hamiltonian’s are typically non-separable, requiring the expensive generation of momenta, with the recent trend being to utilise partial momentum refreshment. Separable Shadow Hamiltonian Hybrid Monte Carlo (S2HMC) employs a canonical transformation which results in the Hamiltonian being separable and makes use of a processed leapfrog integrator. In this work, we combine the benefit of sampling using S2HMC with partial momentum refreshment to create the Separable Shadow Hamiltonian Hybrid Monte Carlo with Partial Momentum Refreshment (PS2HMC) algorithm which leaves the target distribution invariant. Numerical experiments across various targets show that the proposed algorithm outperforms S2HMC and Shadow Hamiltonian Monte Carlo with partial momentum refreshment. Comprehensive analysis is performed on the Banana shaped distribution, multivariate Gaussian distributions of various dimensions, Bayesian logistic regression and Bayesian neural networks.
Highlights
Markov Chain Monte Carlo (MCMC) methods have been successfully employed to sample from complex statistical and machine learning models [1, 2, 3, 4]
The results showed the significant benefits that can be obtained by utilising partial momentum refreshment in Hamiltonian Monte Carlo methods [21, 28]
The performance of the MCMC methods across the various metrics are presented in Figure 3 and Tables 3 to 6
Summary
Markov Chain Monte Carlo (MCMC) methods have been successfully employed to sample from complex statistical and machine learning models [1, 2, 3, 4]. A popular MCMC algorithm is Hamiltonian Monte Carlo (HMC) [9, 17, 18], which utilises Hamiltonian dynamics to sample from the target posterior. HMC improves on random walk samplers such as the Metropolis-Hastings algorithm by utilising the first-order gradient information of the unnormalised posterior distribution to guide its exploration. This results in lower auto-correlations between the generated samples when compared to random walk samplers
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