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 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, sampling occurs in the space of ordered partitions with blocks labelled in the least element order. This improves mixing and promotes a consistent labelling of mixture components throughout iterations. The performance of the proposed algorithm is evaluated on simulated data.