By Nils Lid Hjort, Chris Holmes, Peter Müller, Stephen G. Walker

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Xn ) are exchangeable, the same description applies to any Xi given Xj , j = 1, . . , i − 1, i + 1, . . , n. This procedure, studied in Blackwell and MacQueen (1973), is known as the generalized P´olya urn scheme. This will turn out to have a key role in the development of Markov chain Monte Carlo (MCMC) procedures for latent variables sampled from a Dirichlet process, as in Dirichlet mixtures discussed shortly. Because of ties in the above description, the number of distinct observations, the total number of fresh draws from G including the first, is generally much smaller than n.

Journal of the Royal Statistical Society Series, Series B, 71, 319–72. , Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge: Cambridge University Press. Schwartz, L. (1965). On Bayes procedures. Zeitschrift f¨ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 4, 10–26. Skaug, H. J. and Fournier, D. A. (2006). Automatic approximation of the marginal likelihood in non-Gaussian hierarchical models. Computational Statistics and Data Analysis, 5, 699–709. An invitation to Bayesian nonparametrics 21 Smith, M.

At the risk of being less than perfectly precise, we shall prefer somewhat informal statements and informal arguments leading to these results. An area which we do not attempt to cover is that of Bayesian survival analysis, where several interesting priors have been constructed and consistency and rate of convergence results have been derived. We refer readers to Ghosh and Ramamoorthi (2003) and Ghosal and van der Vaart (2010) as general references for all topics discussed in this chapter. f. is completely arbitrary.

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