  Gibbs sampling methods are standard tools to perform posterior inference for
mixture models. These have been broadly classified into two categories:
marginal and conditional methods. While conditional samplers are more widely
applicable than marginal ones, they may 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, and that can be readily adapted to
mixture models for which marginal samplers are not available. 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-Hastings steps. Additionally, sampling occurs in the space of
ordered partitions with blocks labelled in the least element order, which
endows the sampler with good mixing properties. The performance of the proposed
algorithm is evaluated in a simulation study.
