label.switching (version 1.8)

stephens: 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.

Value

permutations

\(m\times K\) dimensional array of permutations

iterations

integer denoting the number of iterations until convergence

status

returns the exit status

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.

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

# NOT RUN {
#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
# }