Some other references:
https://www.jstor.org/stable/2241268
https://arxiv.org/abs/1308.6306
https://projecteuclid.org/journals/annals-of-mathematical-st...
In the vogue days of reversible jump MCMC I played with mixture estimation of the number of components under a basic prior (an approach which gives decent results in Figs 1 and 3), but I never used a Dirichlet process prior for this problem. This paper points out that even this simple approach is problematic because it’s only consistent if the true distribution is such a mixture, and in my case it definitely was not.
Anyway, one takeaway, esp. from sec 1.2.1, is that the Dirichlet process prior is not suitable for estimating #components in most cases; it favors small clusters. And indeed, the concept of estimating #components is tricky to begin with, as noted above.
Just because you can compute the posterior, doesn’t mean it’s saying what you think it is about the underlying true distribution!