
Componentwise Hit-And-Run Metropolis (CHARM)
Acceptance Rate | The optimal acceptance rate is 44%, and is based on the univariate normality of each marginal posterior distribution. The observed acceptance rate may be suitable in the interval [15%,50%]. |
Applications | This is a widely applicable, general-purpose algorithm. |
Difficulty | This algorithm is easy for a beginner. There are no algorithm specifications (although one is optional), and tuning is unnecessary. |
Final Algorithm? | Yes. |
Proposal | Componentwise. |
The Hit-And-Run algorithm is a variation of RWM that has been around as long as Gibbs sampling. Hit-And-Run randomly samples a direction on the unit sphere, and a proposal is made for each parameter in its randomly-selected direction for a uniformly-distributed distance (Gilks and Roberts, 1996). This version of Hit-And-Run, called Componentwise Hit-And-Run Metropolis (CHARM), includes componentwise proposals and a Metropolis step for rejection. Introduced by Turchin (1971) along with Gibbs sampling, and popularized by Smith (1984), Hit-And-Run was given its name later due to its ability to run across the state-space and arrive at a distant "hit-point". It is related to other algorithms with interesting names, such as Hide-And-Seek and Shake-And-Bake.
CHARM has one optional algorithm specification: alpha.star
. When Specs=NULL
, CHARM is non-adaptive. Otherwise, alpha.star
is the target acceptance rate, and is recommended to be 0.44. When a target acceptance rate is declared, CHARM applies the Robbins-Monro stochastic approximation of Garthwaite et al. (2010) to the proposal distance to attain the target acceptance rate.
As a componentwise algorithm, the model is evaluated after a proposal is made for each parameter, which results in an algorithm that takes more time per iteration. As with the Metropolis-within-Gibbs (MWG) family, the time to complete each iteration grows with the number of parameters. Compared to other algorithms with multivariate proposals, a disadvantage is the time to complete each iteration increases as a function of parameters and model complexity. For example, in a 100-parameter model, CHARM completes its first iteration as HARM completes its 100th.
CHARM enjoys many of the advantages of HARM, such as having no tuning parameters (unless the adaptive form is used), traversing complex spaces with bounded sets in one iteration, not being adaptive (unless specified as adaptive), handling high correlations well, and having the potential to work well with multimodal distributions. When non-adaptive, CHARM may be used as a final algorithm.
References
- Garthwaite P, Fan Y, Sisson S (2010). "Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process."
- Gilks W, Roberts G (1996). "Strategies for Improving MCMC." In W Gilks, S Richardson, D Spiegelhalter (eds.), Markov Chain Monte Carlo in Practice, p. 89-114. Chapman & Hall, Boca Raton, FL.
- Smith R (1984). "Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed Over Bounded Region." Operations Research, 32, 1296-1308.
- Turchin, VF (1971). "On the Computation of Multidimensional Integrals by the Monte Carlo Method", Theory of Probablility and its Applications, 16(4), 720-724.



