Some progress on the analysis of the MAP estimator of partition in standard BNP mixture models

I am going to present progress in the research regarding the maximal a posteriori estimator (MAP) in a given standard Bayesian Non-Parametric (BNP) mixture models. The question originates from [1] where authors investigated the consistency of the posterior probability in a wide range of these models and proved its inconsistency in some sense. I'll start the presentation with a brief overview of the results of [2] and then move to the description of possible generalisations and analogous problems in different settings. This is still an on-going work, so in this part there will be probably more unanswered than answered questions, but I'll presented the justifications of proved statements and perhaps ask the audience for some advice with regard to those problems that I still cannot solve.

[1] Miller, Jeffrey W., and Matthew T. Harrison. "Inconsistency of Pitman-Yor process mixtures for the number of components." The Journal of Machine Learning Research 15.1 (2014): 3333-3370.
[2] Rajkowski, Łukasz. "Analysis of the maximal a posteriori partition in the Gaussian Dirichlet Process Mixture Model." Bayesian Analysis (2018).