EntropyParallel {EntropyMCMC} R Documentation

## Parallel simulation and Entropy estimation of MCMC's - single core and cluster versions

### Description

This function simulates “parallel chains” (iid copies) of a MCMC algorithm, i.e. for each “time” iteration t the next step of all the nmc chains are generated, then the Entropy of the density p^t of the algorithm at iteration t, E_{p^t}[\log(p^t)], and the Kullback divergence between p^t and the target density are estimated, based on these nmc steps iid from p^t. By default keep.all = FALSE i.e. the past of the parallel chains is discarded so that the amount of memory requirement is kept small, and only entropy-related estimates are returned. If keep.all = TRUE, the entire set of chains trajectories is saved in an array of dimensions (n,d,nmc), such as the one returned by MCMCcopies or MCMCcopies.cl.

A version of this function implementing several HPC (parallel) computing strategies is available (see details).

### Usage

EntropyParallel(mcmc_algo, n = 100, nmc = 10, Ptheta0, target, f_param, q_param,
method = "A.Nearest.Neighbor",k=1, trim = 0.02, keep.all = FALSE,
verb = TRUE, EntVect = FALSE)

EntropyParallel.cl(mcmc_algo, n = 100, nmc = 10, Ptheta0, target, f_param, q_param,
method = "A.Nearest.Neighbor",k=1, eps = 0, trim=0.02,
verb=TRUE, EntVect = FALSE, cltype="PAR_SOCK", nbnodes = 4,
par.logf = FALSE, uselogtarget = FALSE, logtarget = NULL)


### 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,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 the example below. f_param A list holding all the necessary target parameters, including the data in an actual Bayesian model, and consistent with the target definition. q_param A list holding all the necessary parameters for the proposal density of the MCMC algorithm mcmc_algo. method The method for estimating the entropy E_{p^t}[\log(p^t)]. The methods currently implemented for this function are "Nearest.Neighbor" as in Kozachenko and Leonenko (1987), "k.Nearest.Neighbor" as in Leonenko et al. (2005) (the default in the single core version), and "A.Nearest.Neighbor" which is as "k.NearestNeighbor" using the RANN package for (Approximate) fast computation of nearest neighbors, instead of R code (the default for the cluster version). Other methods such as "Gyorfi.trim" subsampling method as defined in Gyorfi and Vander Mulen (1989) are available as well (see Chauveau and Vandekerkhove (2012)). k The k-nearest neighbor index, the default is k=1. eps Error bound: default of 0.0 implies exact nearest neighbour search, see the RANN package. trim not used in this implementation, only for method="Gyorfi.trim" keep.all If TRUE, all the simulated chains are stored in a 3-dimensional array of dimensions (n,d,nmc), such as the one returned by MCMCcopies verb Verbose mode for summarizing output during the simulation. EntVect If FALSE (the default), the entropy is computed only on the kth-nearest neighbor. If TRUE, the entropy is computed for all j-NN's for j=1 to k (the latter being mostly for testing purposes). cltype Character string specifying the type of cluster; currently implemented types are: "PAR_SOCK" for socket cluster with parallel library, the default; "SNOW_SOCK" for socket cluster with snow library, and "SNOW_RMPI" for snow MPI cluster with Rmpi library. nbnodes The number of nodes or virtual cores requested to run the nmc simulations in parallel. For the snow version, defaults to all; for the cluster version, defaults to 4. par.logf if TRUE, then the computation of the log of the target density at each of the nmc chain locations, needed for the NN procedure is also executed in parallel using parRapply(). This may speed up the process if the target is complicated i.e. takes some time to evaluate. If the target is simple enough (like target_norm), then communications between nodes are slower than computations, in which case par.logf = FALSE (the default) should be preferred. uselogtarget Set to FALSE by default; useful in some cases where log(f(\theta)) returns -Inf values in Kullback computations because f(\theta) itself returns too small values for some \theta far from modal regions. In these case using a function computing the logarithm of the target can remove the infinity values. logtarget The function defining log(f(theta)), NULL by default, required if uselogtarget equals TRUE. This option and uselogtarget are currently implemented only for the "A.Nearest.Neighbor" method, and for the default EntVect = FALSE option.

### Details

For the HPC (parallel) version, the computation of the nmc chains next step are done by the cluster nodes: EntropyParallel.cl is a generic cluster version implementing several types of cluster for running on a single, multicore computer or on a true cluster using MPI communications. It is under development and may not work on all platform/OS. For instance the parallel socket cluster version does not work on Windows machines (see the parallel package documentation). Currently tested under LINUX, Mac OSX, and a cluster using OpenMPI and Sun Grid Engine.

Note that the parallel computing for this approach is less efficient than the two-steps procedure consisting in (i) parallel simulation of the iid chains using MCMCcopies.cl to generate the “cube” of simulated values, and then (ii) entropy and Kullback estimation using EntropyMCMC.mc. This is because each node computes only one iteration at a time for the nmc chains here, whereas it computes all the n iterations once for the nmc chains when the entire cube is saved first. This is a trade-off between memory and speed.

Note also that the Rmpi option is less efficient than the default option using parallel if you are running on a single computer. MPI communication are required only for running on a true cluster/grid.

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; IID_norm, a gaussian iid sampler which is merely a "fake" MCMC for testing purposes.

Currently only non-adaptive Markov chains or adaptive chains for which the past can be summarized by some sufficient statistics are eligible for this computation forgetting the past of the nmc chains. Adaptive chains such as AMHaario, the Adaptive-Metropolis (AM) from Haario (2001) are not yet tested for this function.

### Value

An object of class "KbMCMC", containing

 Kullback A vector of estimated K(p^t,f), for t=1 up to the number of iterations n. This is the convergence/comparison criterion. Entp A vector of estimated E_{p^t}[\log(p^t)], for t=1 up to the number of iterations that have been simulated. nmc The number of iid copies of each single chain. dim The state space dimension of the MCMC algorithm. algo The name of the MCMC algorithm that have been used to simulate the copies of chains, see MCMCcopies. 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. method The method input parameter (see above). 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 that have been used. prob.accept Estimated rate of acceptation (meaningful for accept/reject-type algorithms). Ptheta The nmc copies of chains in an array(n,d,nmc) of simulated values, where 1st value (1,d,nmc) is Ptheta0.

### Author(s)

Didier Chauveau, Houssam Alrachid.

### 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.

MCMCcopies, MCMCcopies.mc and MCMCcopies.cl for just simulating the iid chains, and EntropyMCMC or EntropyMCMC.mc for computing entropy and Kullback estimates from an already simulated set of iid chains (internally or from external code).

### Examples

## Toy example using the bivariate gaussian target
## same as for MCMCcopies
n = 150; nmc = 50; 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 and entropy + Kullback using the singlecore version
e1 <- EntropyParallel(RWHM, n, nmc, Ptheta0, target_norm,
target_norm_param, q_param, verb = FALSE)
par(mfrow=c(1,2))
plot(e1) # default plot.plMCMC method, convergence after about 80 iterations
plot(e1, Kullback = FALSE) # Plot Entropy estimates over time
abline(normEntropy(target_norm_param), 0, col=8, lty=2) # true E_f[log(f)]

# Another example using multicore version, (not available on Windows)
varq=0.05 # variance of the proposal, even smaller
q_param=list(mean=rep(0,d),v=varq*diag(d))
n=300 # requires more iterations to show convergence
e1 <- EntropyParallel.cl(RWHM, n, nmc, Ptheta0, target_norm,
target_norm_param, q_param, cltype="PAR_SOCK",
verb = FALSE, nbnodes = 2)
plot(e1) # convergence after about 150 iterations



[Package EntropyMCMC version 1.0.4 Index]