Model based labeling for mixture models
Label switching is one of the fundamental problems for Bayesian mixture model analysis. Due to the permutation invariance of the mixture posterior, we can consider that the posterior of a m-component mixture model is a mixture distribution with m! symmetric components and therefore the object of labeling is to recover one of the components. In order to do labeling, we propose to first fit a symmetric m!-component mixture model to the Markov chain Monte Carlo (MCMC) samples and then choose the label for each sample by maximizing the corresponding classification probabilities, which are the probabilities of all possible labels for each sample. Both parametric and semi-parametric ways are proposed to fit the symmetric mixture model for the posterior. Compared to the existing labeling methods, our proposed method aims to approximate the posterior directly and provides the labeling probabilities for all possible labels and thus has a model explanation and theoretical support. In addition, we introduce a situation in which the “ideally” labeled samples are available and thus can be used to compare different labeling methods. We demonstrate the success of our new method in dealing with the label switching problem using two examples.
KeywordsBayesian mixtures Labeling probabilities Label switching Markov chain Monte Carlo Mixture model
Unable to display preview. Download preview PDF.
- Böhning, D.: Computer-Assisted Analysis of Mixtures and Applications. Chapman and Hall/CRC, Boca Raton (1999) Google Scholar
- Celeux, G.: Bayesian inference for mixtures: The label switching problem. In: Payne, R., Green, P.J. (eds.) Compstat 98-Proc. in Computational Statistics, pp. 227–232. Physica, Heidelberg (1998) Google Scholar
- Jasra, A.: Bayesian inference for mixture models via Monte Carlo. Ph.D. Thesis, Imperial College London (2005) Google Scholar
- Marin, J.-M., Mengersen, K.L., Robert, C.P.: Bayesian modelling and inference on mixtures of distributions. In: Dey, D., Rao, C.R. (eds.) Handbook of Statistics 25. North-Holland, Amsterdam (2005) Google Scholar