ecr.iterative.2: ECR algorithm (iterative version 2)
In label.switching: Relabelling MCMC Outputs of Mixture Models

Description Usage Arguments Details Value Author(s) References See Also Examples

This function applies the second iterative version of Equivalence Classes Representatives (ECR) algorithm (Papastamoulis and Iliopoulos, 2010, Rodriguez and Walker, 2012). The set of all allocation variables is partitioned into equivalence classes and exactly one representative is chosen from each class. In this version the m\times n \times K of allocation probabilities should be given as input as well.

1	ecr.iterative.2(z, K, p, threshold, maxiter)

`z`	m\times n integer array of the latent allocation vectors generated from an MCMC algorithm.
`K`	the number of mixture components (at least equal to 2).
`p`	m\times n \times K dimensional array of allocation probabilities of the n observations among the K mixture components, for each iteration t = 1,…,m of the MCMC algorithm.
`threshold`	An (optional) positive number controlling the convergence criterion. Default value: 1e-6.
`maxiter`	An (optional) integer controlling the max number of iterations. Default value: 100.

For a given MCMC iteration t=1,…,m, let w_k^{(t)} and θ_k^{(t)}, k=1,…,K denote the simulated mixture weights and component specific parameters respectively. Then, the (t,i,k) element of p corresponds to the conditional probability that observation i=1,…,n belongs to component k and is proportional to p_{tik} \propto w_k^{(t)} f(x_i|θ_k^{(t)}), k=1,…,K, where f(x_i|θ_k) denotes the density of component k. This means that:

p_{tik} = \frac{w_k^{(t)} f(x_i|θ_k^{(t)})}{w_1^{(t)} f(x_i|θ_1^{(t)})+… + w_K^{(t)} f(x_i|θ_K^{(t)})}.

In case of hidden Markov models, the probabilities w_k should be replaced with the proper left (normalized) eigenvector of the state-transition matrix.

`permutations`	m\times K dimensional array of permutations
`iterations`	integer denoting the number of iterations until convergence
`status`	returns the exit status

Panagiotis Papastamoulis

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.

Rodriguez C.E. and Walker S. (2014). Label Switching in Bayesian Mixture Models: Deterministic relabeling strategies. Journal of Computational and Graphical Statistics. 23:1, 25-45

permute.mcmc, label.switching, ecr, ecr.iterative.1, stephens

#load a toy example: MCMC output consists of the random beta model
# applied to a normal mixture of \code{K=2} components. The number of
# observations is equal to \code{n=5}. The number of MCMC samples is
# equal to \code{m=1000}. The 300 simulated allocations are stored to
# array \code{z}. The matrix of allocation probabilities is stored to
# array \code{p}. 
data("mcmc_output")
z<-data_list$"z"
K<-data_list$"K"
p<-data_list$"p"
# mcmc parameters are stored to array \code{mcmc.pars}
mcmc.pars<-data_list$"mcmc.pars"
# mcmc.pars[,,1]: simulated means of the two components
# mcmc.pars[,,2]: simulated variances 
# mcmc.pars[,,3]: simulated weights
# the relabelling algorithm will run with the default initialization
# (no opt_init is specified)
run<-ecr.iterative.2(z = z, K = 2, p = p)
# apply the permutations returned by typing:
reordered.mcmc<-permute.mcmc(mcmc.pars,run$permutations)
# reordered.mcmc[,,1]: reordered means of the two mixture components
# reordered.mcmc[,,2]: reordered variances of the two components
# reordered.mcmc[,,3]: reordered weights of the two components