Gibbs sampling for mixtures in order of appearance: the ordered allocation sampler

Gibbs sampling methods for mixture models are based on data augmentation schemes that account for the unobserved partition in the data. Conditional samplers rely on allocation variables that identify each observation with a mixture component. They are known to suffer from slow mixing in infinite mixtures, where some form of truncation, either deterministic or random, is required. In mixtures with random number of components, the exploration of parameter spaces of different dimensions can also be challenging. We tackle these issues by expressing the mixture components in the random order of appearance in an exchangeable sequence directed by the mixing distribution. We derive a sampler that is straightforward to implement for mixing distributions with tractable size-biased ordered weights. In infinite mixtures, no form of truncation is necessary. As for finite mixtures with random dimension, a simple updating of the number of components is obtained by a blocking argument, thus, easing challenges found in trans-dimensional moves via Metropolis-Hasting steps. Additionally, the latent clustering structure of the model is encrypted by means of an ordered partition with blocks labelled in the least element order, which mitigates the label-switching problem. We illustrate through a simulation study the good mixing performance of the new sampler.