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Hybrid Monte Carlo on Hilbert spaces

Author

Listed:
  • Beskos, A.
  • Pinski, F.J.
  • Sanz-Serna, J.M.
  • Stuart, A.M.
Abstract
The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures [pi] which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself. We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure [Pi] in an enlarged phase space having the target [pi] as a marginal, together with a Hamiltonian flow that preserves [Pi]; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of [Pi] when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space.

Suggested Citation

  • Beskos, A. & Pinski, F.J. & Sanz-Serna, J.M. & Stuart, A.M., 2011. "Hybrid Monte Carlo on Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2201-2230, October.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2201-2230
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    References listed on IDEAS

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    1. Mark Girolami & Ben Calderhead, 2011. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(2), pages 123-214, March.
    2. Gareth O. Roberts & Jeffrey S. Rosenthal, 1998. "Optimal scaling of discrete approximations to Langevin diffusions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 255-268.
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    Cited by:

    1. Cheng Zhang & Babak Shahbaba & Hongkai Zhao, 2017. "Precomputing strategy for Hamiltonian Monte Carlo method based on regularity in parameter space," Computational Statistics, Springer, vol. 32(1), pages 253-279, March.
    2. Beskos, Alexandros & Kalogeropoulos, Konstantinos & Pazos, Erik, 2013. "Advanced MCMC methods for sampling on diffusion pathspace," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1415-1453.
    3. Simon Byrne & Mark Girolami, 2013. "Geodesic Monte Carlo on Embedded Manifolds," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 825-845, December.
    4. Derek Tucker, J. & Shand, Lyndsay & Chowdhary, Kenny, 2021. "Multimodal Bayesian registration of noisy functions using Hamiltonian Monte Carlo," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).
    5. Simon Byrne & Mark Girolami, 2014. "Rejoinder: Geodesic Monte Carlo on Embedded Manifolds," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 19-21, March.

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