relabelling mixtures (#2)

Following the previous post, I went and had  a (long) look at Puolamäki and Kaski’s paper. I must acknowledge that, despite having several runs through the paper, I still have trouble with the approach… From what I understand, the authors use a Bernoulli mixture pseudo-model to reallocate the observations to components.  That is, given an MCMC output with simulated allocations variables (a.k.a., hidden or latent variables), they create a (TxK)xn matrix of component binary indicators e.g., for a three component mixture,

0 1 0 0 1 0…
1 0 0 0 0 0…
0 0 1 1 0 1…
0 1 0 0 1 1…

and estimate a probability to be in component j for each of the n observations, according to the (pseudo-)likelihood

\prod_{r=1}^R \sum_{j=1}^K \prod_{i=1}^n \beta_{i,j}^{z_{i,r}}(1-\beta_{i,j})^{1-z_{i,r}}

It took me a few days, between morning runs and those wee hours when I cannot get back to sleep (!), to make some sense of this Bernoulli modelling. The allocation vectors are used together to estimate the probabilities of being “in” component j together. However the data—which is the outcome of an MCMC simulation and de facto does not originate from that Bernoulli mixture—does not seem appropriate, both because it is produced by an MCMC simulation and is made of blocks of highly correlated rows [which sum up to one]. The Bernoulli likelihood above also defines a new model, with many more parameters than in the original mixture model. And I fail to see why perfect, partial or inexistent label switching [in the MCMC sequence] is not going to impact the estimation of the Bernoulli mixture. And why an argument based on a fixed parameter value (Theorem 3) extends to an MCMC outcome where parameters themselves are subjected to some degree of label switching. Bemused, I remain…

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