MCMCcopies.mc {EntropyMCMC}

Simulates iid copies of a MCMC algorithm - multicore version

Description

Simulates nmc iid copies of a MCMC algorithm mcmc_algo for n (time) iterations and returns an object of class plMCMC (for parallel MCMC) holding an array of the trajectories and running information. This functions is similar to MCMCcopies except that it uses the parallel package (available in the main distribution, but not for Windows machines) to split the task between the available virtual cores on the computer.

Usage

MCMCcopies.mc(mcmc_algo, n = 100, nmc = 10, Ptheta0, target, f_param, q_param, 
			verb = TRUE, nbcores=detectCores())

Arguments

mcmc_algo

a list defining an MCMC algorithm in terms of the functions it uses, such as RWHM, see details below.

n

The number of (time) iterations of each single chain to run.

nmc

The number of iid copies of each single chain.

Ptheta0

A (nmc x d) matrix, with the ith row giving a d-dimensional initial theta values for the ith chain.

target

The target density for which the MCMC algorithm is defined; may be given only up to a multiplicative constant for most MCMC. target must be a function such as the multidimensional gaussian target_norm(x,param) with argument and parameters passed like in this example.

f_param

A list holding all the necessary target parameters, consistent with the target definition.

q_param

A list holding all the necessary parameters for the proposal density of the MCMC algorithm mcmc_algo.

verb

Verbose mode for summarizing output during the simulation.

nbcores

Number of required (virtual) cores, defaults to all as returned by detectCores().

Details

MCMCcopies.mc, like MCMCcopies, sequentially simulates nmc iid copies of the MCMC algorithm passed in the list mcmc_algo, for n (time) iterations, and returns an object of class plMCMC holding an array of the trajectories and running information. The list mcmc_algo must contain the named elements:

• name, the name of the MCMC, such as "RWHM"

• chain, the function for simulation of n steps of a single chain

• step, the function for simulation of 1 step of that algorithm

• q_pdf, the density of the proposal

• q_proposal, the function that simulates a proposal

For examples, see the algorithms currently implemented: RWHM, the Random Walk Hasting-Metropolis with gaussian proposal; HMIS_norm, an Independence Sampler HM with gaussian proposal; AMHaario, the Adaptive-Metropolis (AM) from Haario (2001); IID_norm, a gaussian iid sampler which is merely a "fake" MCMC for testing purposes.

Value

MCMCcopies returns a list of class plMCMC with items:

Ptheta

The nmc copies of chains in an array(n,d,nmc) of simulated values, where 1st value (1,d,nmc) is Ptheta0.

prob.accept

The estimated rate of acceptation over all simulations.

algo

The MCMC algorithm name i.e. mcmc_algo$name.

target

The target density.

f_param

The list holding all the target parameters.

q_param

The list holding all the proposal density parameters.

Author(s)

Didier Chauveau.

References

• Chauveau, D. and Vandekerkhove, P. (2013), Smoothness of Metropolis-Hastings algorithm and application to entropy estimation. ESAIM: Probability and Statistics, 17, 419–431. DOI: http://dx.doi.org/10.1051/ps/2012004

• Chauveau D. and Vandekerkhove, P. (2014), Simulation Based Nearest Neighbor Entropy Estimation for (Adaptive) MCMC Evaluation, In JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association. 2816–2827.

• Chauveau D. and Vandekerkhove, P. (2014), The Nearest Neighbor entropy estimate: an adequate tool for adaptive MCMC evaluation. Preprint HAL http://hal.archives-ouvertes.fr/hal-01068081.

See Also

A more general cluster version MCMCcopies.cl, a single core version MCMCcopies, and functions doing simulation and entropy and Kullback estimation simultaneously: EntropyParallel and EntropyParallel.cl

Examples

## Toy example using the bivariate gaussian target

## not working on Windows since socket cluster not implemented
n = 150; nmc = 20; d=2 # bivariate example
varq=0.1 # variance of the proposal (chosen too small)
q_param=list(mean=rep(0,d),v=varq*diag(d))
## initial distribution, located in (2,2), "far" from target center (0,0)
Ptheta0 <- DrawInit(nmc, d, initpdf = "rnorm", mean = 2, sd = 1) 
# simulations (may be compared with the singlecore version using system.time)
s1 <- MCMCcopies.mc(RWHM, n, nmc, Ptheta0, target_norm,
                    target_norm_param, q_param, nbcores = 2)
summary(s1) # method for "plMCMC" object
  
## see MCMCcopies example for plots