Bayesian nonparametrics and variance regression-: mixtures of Dirichlet processes and the slice sampler
Seminar Room 1, Newton Institute
Mixtures of Dirichlet Processes (MDP) have been widely used as a method of overcoming the discreteness of the Dirichlet Process (DP). The two approaches taken to sample from the Dirichlet measure are: the marginal approach (Escobar and West 1995) where the measure is integrated out within the Gibbs sampler via a clever use of the Polya Urn construction of the DP and the conditional approach (see Ishwaran and Zarepour 2000,2002) which makes use of the infinite sum construction of the DP (see Sethuraman 1994). The ways around this infinite sum construction are either approximations or truncations (Ishwaran and Zarepour 2000) or using the retrospective sampler (see Papaspiliopoulos and Roberts 2005). The retrospective sampler deals with the infinite sum directly, via use of reversible jump steps. We introduce a simpler sampler, which instead of using reversible jumps, introduces an auxiliary variable and incorporates the slice sampler within the construction of the posteriors for the Gibbs sampler (see P. Damien, J. Wakefield, S.G.Walker 1999). The new algorithm works with the infinite sum construction of the DP from the very beginning and by introducing auxiliary variables the Gibbs sampler updating is done within finite sets.