stephens {label.switching}

Stephens' algorithm

Description

Stephens (2000) developed a relabelling algorithm that makes the permuted sample points to agree as much as possible on the n\times K matrix of classification probabilities, using the Kullback-Leibler divergence. The algorithm's input is the matrix of allocation probabilities for each MCMC iteration.

Usage

stephens(p, threshold, maxiter)

Arguments

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,\ldots,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.

Details

For a given MCMC iteration t=1,\ldots,m, let w_k^{(t)} and \theta_k^{(t)}, k=1,\ldots,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,\ldots,n belongs to component k and is proportional to p_{tik} \propto w_k^{(t)} f(x_i|\theta_k^{(t)}), k=1,\ldots,K, where f(x_i|\theta_k) denotes the density of component k. This means that:

p_{tik} = \frac{w_k^{(t)} f(x_i|\theta_k^{(t)})}{w_1^{(t)} f(x_i|\theta_1^{(t)})+\ldots + w_K^{(t)} f(x_i|\theta_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.

Value

permutations

m\times K dimensional array of permutations

iterations

integer denoting the number of iterations until convergence

status

returns the exit status

Author(s)

Panagiotis Papastamoulis

References

Stephens, M. (2000). Dealing with label Switching in mixture models. Journal of the Royal Statistical Society Series B, 62, 795-809.

See Also

permute.mcmc, label.switching

Examples

#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=300}. The matrix of allocation probabilities 
# is stored to matrix \code{p}. 
data("mcmc_output")
# mcmc parameters are stored to array \code{mcmc.pars}
mcmc.pars<-data_list$"mcmc.pars"
# mcmc.pars[,,1]: simulated means of the two components
# mcmc.pars[,,2]: simulated variances 
# mcmc.pars[,,3]: simulated weights 
# the computed allocation matrix is p
p<-data_list$"p"
run<-stephens(p)
# apply the permutations returned by typing:
reordered.mcmc<-permute.mcmc(mcmc.pars,run$permutations)
# reordered.mcmc[,,1]: reordered means of the components
# reordered.mcmc[,,2]: reordered variances
# reordered.mcmc[,,3]: reordered weights