A paper about Bayesian inference on mixtures was posted on arXiv last week, as of 13 Jan 2025. Fast sampling and model selection for Bayesian mixture models, by M. E. J. Newman is based on the notion that (genuine) parameters of a mixture model can be marginalized out when using conjugate priors. This is something that we pointed out quite a while ago, in a 1999 paper with George and Marty, which was devised in a long ride from Baltimore to Cornell after JSM 1999, and again in the 2002 Series B perfect sampling paper with George, Kerrie and Mike. (Also written in 1999.) And marginal likelihood can furthermore be approximated along this way as discussed in the more recent papers Bayesian Inference on Mixtures of Distributions with Kate, Kerrie & Jean-Michel, as well as Approximating the marginal likelihood in mixture models with Jean-Michel.
“Standard mixture models, as commonly formulated, also suffer from a technical, but important, difficulty: the existence of empty components. In many models (…) the number of observations in a component can be zero. Arguably this is acceptable for a model with a fixed number of components, but when the number of components is a free random variable it causes ambiguity, because a given division of observations into components can be represented in more than one way in the model. For instance, we could divide observations into two components, or we could divide them into three components, one of which is empty. This in turn creates difficulties when estimating the number of components—do we have two components or three?”
A very puzzling perspective, imho, since potentially empty components are inherent to (both finite and infinite) mixture models with connected issues of prohibiting some improper priors (if not all) and non-identifiability, including non-identifiability of the number of empty components (which remains random conditional on the data!), but different numbers of components lead to different models and their comparison is handled straightforwardly by a Bayesian analysis.
The author then proceeds to “prohibit empty components” [as a prior choice ?] as we did in the original (!) Gibbs sampler for mixtures in 1990 (published in 1994 in Series B!), seeking posterior properness, a trick later validated by Larry Wasserman (in again 1999, the year of mixtures!). Who called the construct the combination of a fixed prior and of a pseudo-likelihood, correctly imho (as the data dependent part is not properly normalised by a function of the parameters), rather than a prior choice. (The very one who stated that “mixtures, like tequila, are evil and should be avoided“.)
From there, the modelling is rather standard, with an arbitrary prior on k, number of components, a random partition model that prohibits empty components, even though the constraint could be more stringent depending on the number of parameters of a given component and the degree of improperness of the prior, as in our 1990 Series B paper. (Impropriety is not discussed in the paper.) Bayesian inference on k is based on the simulated (pseudo-)posterior. The choice therein as the estimated clustering is the most frequent partition (consensus clustering), connected to our proposal of (again!) 1999 with Merrilee and Gilles. While the estimated mixture is not explicited. The approach is assessed as running at an O(k) cost, with no parallel in terms of the data size n, even though the examples include a 59,946 dataset. One notable algorithmic trick when moving k is in selecting a component at random first rather than an observation index.
Some minor issues: detailed balance indicated as required for convergence (p14), label switching is called component switching (p5), higher acceptance rate indicated as meaning improved performances (p7)
Andrew Wilson 
