MCMCcopies function

Simulates iid copies of a MCMC algorithm

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.

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

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.

Details

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.

Returns

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.

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.

Author(s)

Didier Chauveau.

See Also

Two multicore and cluster version MCMCcopies.mc and MCMCcopies.cl, and functions doing simulation and entropy and Kullback estimation simultaneously: EntropyParallel and EntropyParallel.cl

Examples

## Toy example using the bivariate gaussian target
## with default parameters value, see target_norm_param
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) 
# simulation
s1 <- MCMCcopies(RWHM, n, nmc, Ptheta0, target_norm,
                 target_norm_param, q_param, verb = FALSE)
summary(s1) # method for "plMCMC" object
par(mfrow=c(1,2))
plot(s1) # just a path of the iid chains, method for "plMCMC" object
hist(s1$Ptheta[,1,], col=8) # marginal 1