This paper addresses one simple but potentially very important point: That Dirichlet process mixture models can be inconsistent in the number of mixture components that they infer. This is important because DPs are nowadays widely used in various types of statistical modeling, for example when building clustering type algorithms. This can have real-world implications, for example when clustering breast cancer data with the aim of identifying distinct disease subtypes. Such subtypes are used in clinical practice to inform treatment, so identifying the correct number of clusters (and hence subtypes) has a very important real-world impact.
The paper focuses on proofs concerning two specific cases where the DP turns out to be inconsistent. Both consider the case of the "standard normal DPM", where the likelihood is a univariate normal distribution with unit variance, the mean of which is subject to a normal prior with unit variance. The first proof shows that, if the data are drawn i.i.d. from a zero-mean, unit-variance normal (hence matching the assumed DPM model), $P(T=1 | \text{data})$ does not converge to 1. The second proof takes this further, demonstrating that in fact$ P(T=1 | \text{data}) -> 0 $