Abstract
Hamiltonian Monte Carlo is a Markov Chain Monte Carlo method that has been widely applied to numerous posterior inference problems within the machine learning literature. Markov Chain Monte Carlo estimators have higher variance than classical Monte Carlo estimators due to autocorrelations present between the generated samples. In this work we present three new methods for tackling the high variance problem in Hamiltonian Monte Carlo based estimators: 1) We combine antithetic and importance sampling techniques where the importance sampler is based on sampling from a modified or shadow Hamiltonian using Separable Shadow Hamiltonian Hybrid Monte Carlo, 2) We present the antithetic Magnetic Hamiltonian Monte Carlo algorithm that is based on performing antithetic sampling on the Magnetic Hamiltonian Monte Carlo algorithm and 3) We propose the antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithm based on performing antithetic sampling on the Magnetic Momentum Hamiltonian Monte Carlo method. We find that the antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithms produce effective sample sizes that are higher than antithetic Hamiltonian Monte Carlo on all the benchmark datasets. We further find that antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Hamiltonian Monte Carlo produce higher effective sample sizes normalised by execution time in higher dimensions than antithetic Hamiltonian Monte Carlo. In addition, the antithetic versions of all the algorithms have higher effective sample sizes than their non-antithetic counterparts, indicating the usefulness of adding antithetic sampling to Markov Chain Monte Carlo algorithms. The methods are assessed on benchmark datasets using Bayesian logistic regression and Bayesian neural network models.
Highlights
Predictive models of high parameter dimensionality have become the mainstay across a multitude of critical tasks such as medicine, law enforcement, finance and self-driving automobiles [1]–[5]
We present three new antithetic algorithms being antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo, antithetic Magnetic Hamiltonian Monte Carlo and the antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithms
When calculating the predictive performance metrics for S2HMC and A-S2HMC, the results are weighted by the importance weights in equation (19)
Summary
Predictive models of high parameter dimensionality have become the mainstay across a multitude of critical tasks such as medicine, law enforcement, finance and self-driving automobiles [1]–[5]. HMC serves as an improvement to other MCMC methods, like other MCMC methods it still suffers from the presence of autocorrelations in the generated samples [10], [11] This results in the high variance of HMC based estimators. In this paper we present methods that use the results from coupling theory to create anti-correlated HMC based chains where the momentum variable is shared between the chains. These chains share the random uniform variable in the Metropolis-Hastings acceptance step When these two chains anti-couple strongly, taking the average of the two chains results in estimators that have lower variance, or equivalently, these chains produce samples that have higher effective sample sizes than their non-antithetic counterparts.
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