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Register a new userFaster Hamiltonian Monte Carlo by Learning Leapfrog Scale
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Authors
Changye Wu
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Hamiltonian Monte Carlo samplers have become standard algorithms for MCMC implementations, as opposed to more bas
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We present a method for performing Hamiltonian Monte Carlo that largely eliminates sample rejection for typical hyperparameters. In situations that would normally lead to rejection, instead a longer trajectory is computed until a new state is reached that can be accepted. This is achieved using Markov chain transitions that satisfy the fixed point equation, but do not satisfy detailed balance. The resulting algorithm significantly suppresses the random walk behavior and wasted function evaluations that are typically the consequence of update rejection. We demonstrate a greater than factor of two improvement in mixing time on three test problems. We release the source code as Python and MATLAB packages.
Hamiltonian Monte Carlo (HMC) has been widely adopted in the statistics community because of its ability to sample high-dimensional distributions much more efficiently than other Metropolis-based methods. Despite this, HMC often performs sub-optimally on distributions with high correlations or marginal variances on multiple scales because the resulting stiffness forces the leapfrog integrator in HMC to take an unreasonably small stepsize. We provide intuition as well as a formal analysis showing how these multiscale distributions limit the stepsize of leapfrog and we show how the implicit midpoint method can be used, together with Newton-Krylov iteration, to circumvent this limitation and achieve major efficiency gains. Furthermore, we offer practical guidelines for when to choose between implicit midpoint and leapfrog and what stepsize to use for each method, depending on the distribution being sampled. Unlike previous modifications to HMC, our method is generally applicable to highly non-Gaussian distributions exhibiting multiple scales. We illustrate how our method can provide a dramatic speedup over leapfrog in the context of the No-U-Turn sampler (NUTS) applied to several examples.
Continuous time Hamiltonian Monte Carlo is introduced, as a powerful alternative to Markov chain Monte Carlo methods for continuous target distributions. The method is constructed in two steps: First Hamiltonian dynamics are chosen as the deterministic dynamics in a continuous time piecewise deterministic Markov process. Under very mild restrictions, such a process will have the desired target distribution as an invariant distribution. Secondly, the numerical implementation of such processes, based on adaptive numerical integration of second order ordinary differential equations is considered. The numerical implementation yields an approximate, yet highly robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of the complete Hamiltonian trajectories (hence the title). The proposed algorithm may yield large speedups and improvements in stability relative to relevant benchmarks, while incurring numerical errors that are negligible relative to the overall Monte Carlo errors.
Dynamically rescaled Hamiltonian Monte Carlo (DRHMC) is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterisation so that the re-parameterised target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterisations of the conditional distributions specifying the hierarchical model are constant information parameterisations (CIP), the relation between the modified- and original parameterisation is bijective, explicitly computed and admit exploitation of sparsity in the numerical linear algebra involved. CIPs for a large catalogue of statistical models are presented, and from the catalogue, it is clear that many CIPs are currently routinely used in statistical computing. A relation between the proposed methodology and a class of explicitly integrated Riemann manifold Hamiltonian Monte Carlo methods is discussed. The methodology is illustrated on several example models, including a model for inflation rates with multiple levels of non-linearly dependent latent variables.
In this paper, we develop Bayesian Hamiltonian Monte Carlo methods for inference in asymmetric GARCH models under different distributions for the error term. We implemented Zero-variance and Hamiltonian Monte Carlo schemes for parameter estimation to try and reduce the standard errors of the estimates thus obtaing more efficient results at the price of a small extra computational cost.
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