• Corpus ID: 237346904

An Introduction to Hamiltonian Monte Carlo Method for Sampling

@article{Vishnoi2021AnIT,
  title={An Introduction to Hamiltonian Monte Carlo Method for Sampling},
  author={Nisheeth K. Vishnoi},
  journal={ArXiv},
  year={2021},
  volume={abs/2108.12107}
}
The goal of this article is to introduce the Hamiltonian Monte Carlo method – a Hamiltonian dynamics inspired algorithm for sampling from a Gibbs density π(x) ∝ e−f(x). We focus on the “idealized” case, where one can compute continuous trajectories exactly. We show that idealized HMC preserves π and we establish its convergence when f is strongly convex and smooth. 
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References

SHOWING 1-10 OF 64 REFERENCES

On the convergence of Hamiltonian Monte Carlo

This paper discusses the irreducibility and geometric ergodicity of the Hamiltonian Monte Carlo (HMC) algorithm and provides verifiable conditions on $\F$ under which the HMC sampler is geometrically ergodic.

Positive Curvature and Hamiltonian Monte Carlo

It is shown that positive curvature can be used to prove theoretical concentration results for HMC Markov chains and to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian.

On the geometric ergodicity of Hamiltonian Monte Carlo

We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both…

Hybrid Monte Carlo

Stochastic Gradient Hamiltonian Monte Carlo

A variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution is introduced.

Hamiltonian Monte Carlo for Hierarchical Models

This paper explores the use of Hamiltonian Monte Carlo for hierarchical models and demonstrates how the algorithm can overcome those pathologies in practical applications.

Convergence rate of Riemannian Hamiltonian Monte Carlo and faster polytope volume computation

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of…

MCMC Using Hamiltonian Dynamics

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of…

Randomized Hamiltonian Monte Carlo

Tuning the durations of the Hamiltonian flow in Hamiltonian Monte Carlo (also called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational cost and sampling quality, which is typically…

Optimal scaling of discrete approximations to Langevin diffusions

An asymptotic diffusion limit theorem is proved and it is shown that, as a function of dimension n, the complexity of the algorithm is O(n1/3), which compares favourably with the O- complexity of random walk Metropolis algorithms.
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