Phoenix

Tempered Hamiltonian Monte Carlo (THMC)


Aspect
Description
Acceptance Rate The optimal acceptance rate is 65% when L > 1, or 57.4% when L = 1. The observed acceptance rate may be suitable in the interval [60%,70%] when L > 1, or [40%,80%] when L = 1.
Applications This is a widely applicable, general-purpose algorithm that is best suited to models with a small number of parameters. The number of model evaluations per iteration increases with the number of parameters.
Difficulty This algorithm is difficult for a beginner. It has a several algorithm specifications, and these are difficult to tune.
Final Algorithm? Yes.
Proposal Multivariate. Proposals are multivariate only in the sense that proposals for multiple parameters are generated at once. Each iteration involves numerous proposals, due to partial derivatives and L.

The Tempered Hamiltonian Monte Carlo (THMC) algorithm is an extension of Hamiltonian Monte Carlo (HMC) to include another algorithm specification: Temperature, which must be positive. When greater than 1, the algorithm should explore more diffuse distributions, and may be helpful with multimodal distributions.

THMC has four algorithm specifications: step-size epsilon, the number L of leapfrog steps, an optional mass matrix M, and Temperature. Algorithm specifications are the same as for HMC, with the exception of temperature, which is described below.

There are a variety of ways to include tempering in HMC, and this algorithm, named here as THMC, uses "tempered trajectory", as described by Neal (2011). When L > 1 and during the first half of the leapfrog steps, the momentum is increased (heated) by multiplying it by √T , where T is temperature, each leapfrog step. In the last half of the leapfrog steps, the momentum decreases (is cooled down) by dividing it by √T. The momentum is largest in the middle of the leapfrog steps, where mode-switching behavior becomes most likely to occur. This preserves the trajectory, εL.

As with HMC, THMC is a difficult algorithm to tune. Since THMC is non-adaptive, it is sufficient as a final algorithm.

References

  • Neal R (2011). "MCMC for Using Hamiltonian Dynamics." In S Brooks, A Gelman, G Jones, M Xiao-Li (eds.), Handbook of Markov Chain Monte Carlo, p. 113-162. Chapman & Hall, Boca Raton, FL.

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