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We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter α. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (2008) and implemented by Yau et al (2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on α. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples.
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