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I am going to study clustering methods in the Bayesian perspective. I understood how k-means works, and I found it pretty clear, due to the notion of distance and assignments to specific centers.

I am now turning the attention to the probabilistic approach and the use of mcmc methods.

I have read this question:

Dirichlet Processes for clustering: how to deal with labels?

and a useful page:

https://snippyhollow.github.io/blog/2013/03/10/collapsed-gibbs-sampling-for-dirichlet-process-gaussian-mixture-models/

However, I do not understand how to assign points to clusters. Which kind of posterior distribution should I compute? What is the meaning of sampling from the posterior distribution in layman's terms? Do I obtain a probability for each point to belong to any possible cluster?

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