SM.MAP.MixReparametrized: summary of the output produced by K.MixReparametrized

Description

Label switching in a simulated Markov chain produced by K.MixReparametrized is removed by the technique of Marin et al. (2004). Namely, component labels are reorded by the shortest Euclidian distance between a posterior sample and the maximum a posteriori (MAP) estimate. Let $\theta_i$ be the $i$-th vector of computed component means, standard deviations and weights. The MAP estimate is derived from the MCMC sequence and denoted by $\theta_{MAP}$. For a permutation $\tau \in \Im_k$ the labelling of $\theta_i$ is reordered by $$\tilde{\theta}_i=\tau_i(\theta_i)$$ where $\tau_i=\arg \min_{\tau \in \Im_k} \mid \mid \tau(\theta_i)-\theta_{MAP}\mid \mid$.

Angular parameters $\xi_1^{(i)}, \ldots, \xi_{k-1}^{(i)}$ and $\varpi_1^{(i)}, \ldots, \varpi_{k-2}^{(i)}$s are derived from $\tilde{\theta}_i$. There exists an unique solution in $\varpi_1^{(i)}, \ldots, \varpi_{k-2}^{(i)}$ while there are multiple solutions in $\xi^{(i)}$ due to the symmetry of $\mid\cos(\xi) \mid$ and $\mid\sin(\xi) \mid$. The output of $\xi_1^{(i)}, \ldots, \xi_{k-1}^{(i)}$ only includes angles on $[-\pi, \pi]$.

The label of components of $\theta_i$ (before the above transform) is defined by

$$\tau_i^*=\arg \min_{\tau \in \Im_k}\mid \mid \theta_i-\tau(\theta_{MAP}) \mid \mid.$$ The number of label switching occurrences is defined by the number of changes in $\tau^*$.

Usage

SM.MAP.MixReparametrized(estimate, xobs, alpha0, alpha)

Arguments

estimate

Output of K.MixReparametrized

xobs

Data set

alpha0

Hyperparameter of Dirichlet prior distribution of the mixture model weights

alpha

Hyperparameter of beta prior distribution of the radial coordinate

Value

MUMatrix of MCMC samples of the component means of the mixture model
SIGMAMatrix of MCMC samples of the component standard deviations of the mixture model
PMatrix of MCMC samples of the component weights of the mixture model
Ang_SIGMAMatrix of computed $\xi$'s corresponding to SIGMA
Ang_MUMatrix of computed $\varpi$'s corresponding to MU. This output only appears when $k > 2$.
Global_MeanMean, median and $95%$ credible interval for the global mean parameter
Global_StdMean, median and $95%$ credible interval for the global standard deviation parameter
PhiMean, median and $95%$ credible interval for the radius parameter
component_muMean, median and $95%$ credible interval of MU
component_sigmaMean, median and $95%$ credible interval of SIGMA
component_pMean, median and $95%$ credible interval of P
l_stayNumber of MCMC iterations between changes in labelling
n_switchNumber of label switching occurrences

Details

Details.

References

Marin, J.-M., Mengersen, K. and Robert, C. P. (2004) Bayesian Modelling and Inference on Mixtures of Distributions, Handbook of Statistics, Elsevier, Volume 25, Pages 459--507.

Examples

Run this code

data(faithful)
xobs=faithful[50:100,1]
#estimate=K.MixReparametrized(xobs,k=2,alpha0=0.5,alpha=0.5,Nsim=1e4)
#result=SM.MAP.MixReparametrized(estimate,xobs,alpha0=0.5,alpha=0.5)

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