🤖 AI Summary
This study investigates the posterior asymptotic behavior of Dirichlet process mixture models when the true data-generating distribution is a finite mixture with $K$ components. By integrating the stick-breaking representation, Wasserstein distance, and posterior contraction theory, the work demonstrates that the model adaptively recovers the true number of components $K$ at the parametric $n^{-1/2}$ rate. Moreover, it reveals a phase transition phenomenon: when estimation accuracy exceeds the $n^{-1/2}$ threshold, the required number of mixture components grows on the order of $\log n$. The analysis further establishes that only $O(\log n)$ components are sufficient to replicate the full posterior’s clustering structure, and any truncated model with at least $K$ components simultaneously achieves optimal posterior contraction rates for both density estimation and the mixing measure.
📝 Abstract
We study the asymptotic properties of the posterior on the latent space for infinite mixtures driven by a Dirichlet process, both in terms of mixing measure and clustering behaviour. In the well-specified regime, where the data are generated by a finite mixture of location densities, we show that the posterior is adaptive to the true number of components $K$: indeed the cumulative mass assigned to weights of the stick-breaking representation beyond the $K$-th one vanishes as $n^{-1/2}$, up to terms growing slower than any polynomial. This also implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. A remarkable phase transition underlies this result: approximating the mixing measure to any precision finer than $n^{-1/2}$ requires a number of components growing logarithmically with the sample size. We show that this has a profound impact on the clustering behaviour: the number of clusters grows logarithmically, as in the prior case, but the proportion of observations outside the $K$ largest clusters vanishes polynomially fast. Finally, we turn these results into posterior guarantees for truncation-based approximations: while any truncation with at least $K$ elements recovers the optimal contraction rates for both density and mixing measure, $\mathcal{O}(\log n)$ components are both necessary and sufficient to reproduce the clustering of the exact posterior.