Bayesian Estimation of Mixtures of Multivariate Bernoulli Distributions

Description

Fully Bayesian inference for estimating the number of clusters and related parameters to heterogeneous binary data.

Details

This package can be used in order to cluster multivariate binary data (NAs are allowed). The main function of the package is coupledMetropolis.

The input is an n \times d binary array where n and d denote the number of observations and dimension of the data. The underlying model is a mixture of independent multivariate Bernoulli distributions with an unknown number of components:

x_i\sim\sum_{k=1}^{K}\pi_k\prod_{j=1}^{d}f(x_{ij};\theta_{kj}),

with x_i = (x_{i1},\ldots,x_{id}); d>1, independent for i = 1,\ldots,n. The term f(x_{ij};\theta_{kj}) denotes the probability density function of the Bernoulli distribution with parameter \theta_{kj}\in(0,1). The number of clusters K is a random variable with support \{1,\ldots,K_{\mbox{max}}\}, where K_{max} is an upper bound for the number of clusters. The model uses the following prior assumptions:

K\sim \mbox{discrete}\{1,\ldots,K_{\mbox{max}}\}

(\pi_1,\ldots,\pi_K)|K \sim \mbox{Dirichlet}(\gamma,\ldots,\gamma)

\theta_{kj}|K \sim \mbox{Beta}(\alpha,\beta);\quad \mbox{independent for}\quad k = 1,\ldots,K; j =1,\ldots,d.

The discrete distribution on the number of clusters it can be a truncated Poisson(1) or Uniform distribution. The model uses data augmentation by also considering the (latent) allocation variable z_i, which a priori assigns observation i to cluster k = 1,\ldots,K with probability P(z_i = k|K, \pi_1,\ldots,\pi_K) = \pi_k independently for i=1,\ldots,n.

In order to infer the parameters of the model, a Markov chain Monte Carlo (MCMC) approach is adopted. Given K, the component-specific parameters \pi_k and \theta_{kj} are integrated out and a collapsed allocation sampler which also updates the number of clusters (Nobile and Fearnside, 2007) is implemented. In the case that the observed data contains missing values, the algorithm simulates their values from the corresponding full conditional distribution. In order to improve the mixing of the simulated chain, a Metropolis-coupled MCMC sampler (Altekar et al., 2004) is incorporated. In particular, various heated chains are run in parallel and swaps are proposed between pairs of chains. The number of chains should be equal to the number of available cores. Each chain runs in parallel using the packages foreach and doParallel.

After inferring the most probable number of clusters, the simulated parameters which correspond to this specific value of K are post-processed in order to undo the label switching problem. For this purpose the label.switching package (Papastamoulis, 2016; see also Papastamoulis and Iliopoulos 2010, 2013 and Papastamoulis, 2014) is used.

Author(s)

Panagiotis Papastamoulis

Maintainer: Panagiotis Papastamoulis

References

Altekar G, Dwarkadas S, Huelsenbeck JP, Ronquist F. (2004): Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference. Bioinformatics 20(3): 407-415.

Nobile A and Fearnside A (2007): Bayesian finite mixtures with an unknown number of components: The allocation sampler. Statistics and Computing, 17(2): 147-162.

Papastamoulis P. and Iliopoulos G. (2010). An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. Journal of Computational and Graphical Statistics, 19: 313-331.

Papastamoulis P. and Iliopoulos G. (2013). On the convergence rate of Random Permutation Sampler and ECR algorithm in missing data models. Methodology and Computing in Applied Probability, 15(2): 293-304.

Papastamoulis P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics, Simulation and Computation, 43(4): 913-927.

Papastamoulis P (2016): label.switching: An R package for dealing with the label switching problem in MCMC outputs. Journal of Statistical Software, 69(1): 1-24.

See Also

coupledMetropolis

The allocation sampler algorithm.

Description

This function implements the collapsed allocation sampler of Nobile and Fearnside (2007) at the context of mixtures of multivariate Bernoulli distributions.

Usage

allocationSamplerBinMix(Kmax, alpha, beta, gamma, m, burn, data, 
thinning, z.true, ClusterPrior, ejectionAlpha, Kstart, outputDir, 
metropolisMoves, reorderModels, heat, zStart, LS, rsX, originalX, printProgress)

Arguments

Details

The output is reordered according to the following label-switching solving algorithms: ECR, ECR-ITERATIVE-1 and STEPHENS. In most cases the results of these different algorithms are identical.

Note

This function is recursively called inside the coupledMetropolis function. There is no need to call it separately.

Author(s)

Panagiotis Papastamoulis

References

Nobile A and Fearnside A (2007): Bayesian finite mixtures with an unknown number of components: The allocation sampler. Statistics and Computing, 17(2): 147-162.

Papastamoulis P. and Iliopoulos G. (2010). An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. Journal of Computational and Graphical Statistics, 19: 313-331.

Papastamoulis P. and Iliopoulos G. (2013). On the convergence rate of Random Permutation Sampler and ECR algorithm in missing data models. Methodology and Computing in Applied Probability, 15(2): 293-304.

Papastamoulis P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics, Simulation and Computation, 43(4): 913-927.

Papastamoulis P (2016): label.switching: An R package for dealing with the label switching problem in MCMC outputs. Journal of Statistical Software, 69(1): 1-24.

See Also

coupledMetropolis

collapsed Gibbs sampler

Description

This function applied collapsed Gibbs sampling assuming that the number of mixture components is known.

Usage

collapsedGibbsBinMix(alpha, beta, gamma, K, m, burn, 
	data, thinning, z.true, outputDir)

Arguments

Note

Not really used.

Author(s)

Panagiotis Papastamoulis

complete log-likelihood

Description

Returns the complete log-likelihood of the mixture.

Usage

complete.loglikelihood(x, z, pars)

Arguments

Value

Complete log-likelihood value.

Author(s)

Panagiotis Papastamoulis

Metropolis-coupled Markov chain Monte Carlo sampler

Description

Main function of the package. The algorithm consists of the allocation sampler combined with a MC3 scheme.

Usage

coupledMetropolis(Kmax, nChains, heats, binaryData, outPrefix, 
	ClusterPrior, m, alpha, beta, gamma, z.true, ejectionAlpha, burn)

Arguments

Details

In the case that the most probable number of clusters is larger than 1, the output is post-processed using the label.switching package. In addition to the objects returned to the user (see value below), the complete output of the sampler is written to the directory outPrefix. It consists of the following files:

• K.allChains.txt m\timesnChains matrix containing the simulated values of the number of clusters (K) per chain.

• K.txt the m simulated values of the number of clusters (K) of the cold chain (posterior distribution).

• p.varK.txt the simulated values of the mixture weights (not identifiable).

• rawMCMC.mapK.KVALUE.txt the raw MCMC output which corresponds to the most probable model (not identifiable).

• reorderedMCMC-ECR-ITERATIVE1.mapK.KVALUE.txt the reordered MCMC output which corresponds to the most probable model, reordered according to the ECR-ITERATIVE-1 algorithm.

• reorderedMCMC-ECR.mapK.KVALUE.txt the reordered MCMC output which corresponds to the most probable model, reordered according to the ECR algorithm.

• reorderedMCMC-STEPHENS.mapK.KVALUE.txt the reordered MCMC output which corresponds to the most probable model, reordered according to the STEPHENS algorithm.

• reorderedSingleBestClusterings.mapK.KVALUE.txt the most probable allocation of each observation after reordering the MCMC sample which corresponds to the most probable number of clusters.

• theta.varK.txt the simulated values of Bernoulli parameters (not identifiable).

• z-ECR-ITERATIVE1.mapK.KVALUE.txt the reordered simulated latent allocations which corresponds to the most probable model, reordered according to the ECR-ITERATIVE-1 algorithm.

• z-ECR.mapK.KVALUE.txt the reordered simulated latent allocations which corresponds to the most probable model, reordered according to the ECR algorithm.

• z-KL.mapK.KVALUE.txt the reordered simulated latent allocations which corresponds to the most probable model, reordered according to the STEPHENS algorithm.

• z.varK.txt the simulated latent allocations (not identifiable).

• classificationProbabilities.mapK.KVALUE.csv the reordered classification probabilities per observation after reordering the most probable number of clusters with the ECR algorithm.

• xEstimated.txt Observed data with missing values estimated by their posterior mean estimate. This file is produced only in the case that the observed data contains missing values.

KVALUE will be equal to the inferred number of clusters. Note that the label switching part is omitted in case that the most probable number of clusters is equal to 1.

Value

The basic output of the sampler is returned to the following R objects:

Author(s)

Panagiotis Papastamoulis

References

Altekar G, Dwarkadas S, Huelsenbeck JP, Ronquist F. (2004): Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference. Bioinformatics 20(3): 407-415.

Nobile A and Fearnside A (2007): Bayesian finite mixtures with an unknown number of components: The allocation sampler. Statistics and Computing, 17(2): 147-162.

Papastamoulis P. and Iliopoulos G. (2010). An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. Journal of Computational and Graphical Statistics, 19: 313-331.

Papastamoulis P. and Iliopoulos G. (2013). On the convergence rate of Random Permutation Sampler and ECR algorithm in missing data models. Methodology and Computing in Applied Probability, 15(2): 293-304.

Papastamoulis P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics, Simulation and Computation, 43(4): 913-927.

Papastamoulis P (2016): label.switching: An R package for dealing with the label switching problem in MCMC outputs. Journal of Statistical Software, 69(1): 1-24.

Examples

#generate dataset from a mixture of 2 ten-dimensional Bernoulli distributions.
set.seed(1)
d <- 10 # number of columns
n <- 50 # number of rows (sample size)
K <- 2 	 # true number of clusters
p.true <- myDirichlet(rep(10,K)) # true weight of each cluster
z.true <- numeric(n) # true cluster membership
z.true <- sample(K,n,replace=TRUE,prob = p.true)
#true probability of positive responses per cluster:
theta.true <- array(data = NA, dim = c(K,d)) 
for(j in 1:d){
    theta.true[,j] <- rbeta(K, shape1 = 1, shape2 = 1)
}
x <- array(data=NA,dim = c(n,d)) # data: n X d array
for(k in 1:K){
        myIndex <- which(z.true == k)
        for (j in 1:d){
                x[myIndex,j] <- rbinom(n = length(myIndex), 
			size = 1, prob = theta.true[k,j])   
        }
}
#	number of heated paralled chains
nChains <- 2
heats <- seq(1,0.8,length = nChains)
## Not run: 
cm <- coupledMetropolis(Kmax = 10,nChains = nChains,heats =  heats,
	binaryData = x, outPrefix = 'BayesBinMixExample',
	ClusterPrior = 'poisson', m = 1100, burn = 100)
#	print summary using:
print(cm)

## End(Not run)
# it is also advised to use z.true = z.true in order to directly compare with 
# the true values. In general it is advised to use at least 4 chains with 
#	heats <- seq(1,0.3,length = nChains)

Label switching algorithms

Description

This is a wrapper for the label.switching package. It is used to post-process the generated MCMC sample in order to undo the label switching problem. This function is called internally to the coupledMetropolis command.

Usage

dealWithLabelSwitching(outDir, reorderModels, binaryData, z.true)

Arguments

Details

See the label.switching package.

Author(s)

Panagiotis Papastamoulis

References

Papastamoulis P. and Iliopoulos G. (2010). An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. Journal of Computational and Graphical Statistics, 19: 313-331.

Papastamoulis P. and Iliopoulos G. (2013). On the convergence rate of Random Permutation Sampler and ECR algorithm in missing data models. Methodology and Computing in Applied Probability, 15(2): 293-304.

Papastamoulis P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics, Simulation and Computation, 43(4): 913-927.

Papastamoulis P (2016): label.switching: An R package for dealing with the label switching problem in MCMC outputs. Journal of Statistical Software, 69(1): 1-24.

Label switching algorithms for the case of missing data

Description

This is a wrapper for the label.switching package. It is used to post-process the generated MCMC sample in order to undo the label switching problem. This function is called internally to the coupledMetropolis command.

Usage

dealWithLabelSwitchingMissing(outDir, reorderModels, binaryData, z.true)

Arguments

Details

See the label.switching package.

Author(s)

Panagiotis Papastamoulis

References

Papastamoulis P. and Iliopoulos G. (2010). An artificial allocations based solution to the label switching problem in Bayesian analysis of mixtures of distributions. Journal of Computational and Graphical Statistics, 19: 313-331.

Papastamoulis P. and Iliopoulos G. (2013). On the convergence rate of Random Permutation Sampler and ECR algorithm in missing data models. Methodology and Computing in Applied Probability, 15(2): 293-304.

Papastamoulis P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics, Simulation and Computation, 43(4): 913-927.

Papastamoulis P (2016): label.switching: An R package for dealing with the label switching problem in MCMC outputs. Journal of Statistical Software, 69(1): 1-24.

Standard Gibbs sampler

Description

This function implements full Gibbs sampling to simulate an MCMC sample from the posterior distribution assuming known number of mixture components.

Usage

gibbsBinMix(alpha, beta, gamma, K, m, burn, data, 
	thinning, z.true, outputDir)

Arguments

Details

Not really used.

Author(s)

Panagiotis Papastamoulis

Simulate from Dirichlet distribution

Description

This function simulates random vectors from a Dirichlet distribution.

Usage

myDirichlet(alpha)

Arguments

Author(s)

Panagiotis Papastamoulis

Print function

Description

This function prints a summary of objects returned by the coupledMetropolis function.

Usage

## S3 method for class 'bbm.object'
print(x, printSubset, ...)

Arguments

Details

The function prints the estimated distribution of the number of clusters, the estimated number of observations assigned to each cluster after post-processing the output with three label switching algorithms, as well as the header of the posterior mean estimates of \theta_{kj} (probability of success for cluster k and feature j) (conditionally on the most probable number of clusters).

Author(s)

Panagiotis Papastamoulis

An equation to solve

Description

Approximately solve the equation (25) of Nobile and Fearnside (2007).

Usage

toSolve(a, n, p0)

Arguments

Author(s)

Panagiotis Papastamoulis