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This article constructs a class of random probability measures based onexponentially and polynomially tilting operated on the laws of completelyrandom measures. The class is proved to be conjugate in that it covers bothprior and posterior random probability measures in the Bayesian sense.Moreover, the class includes some common and widely used random probabilitymeasures, the normalized completely random measures (James (Poisson processpartition calculus with applications to exchangeable models and Bayesiannonparametrics (2002) Preprint), Regazzini, Lijoi and Pr\"{u}nster (Ann.Statist. 31 (2003) 560-585), Lijoi, Mena and Pr\"{u}nster (J. Amer. Statist.Assoc. 100 (2005) 1278-1291)) and the Poisson-Dirichlet process (Pitman and Yor(Ann. Probab. 25 (1997) 855-900), Ishwaran and James (J. Amer. Statist. Assoc.96 (2001) 161-173), Pitman (In Science and Statistics: A Festschrift for TerrySpeed (2003) 1-34 IMS)), in a single construction. We describe an augmentedversion of the Blackwell-MacQueen P\'{o}lya urn sampling scheme (Blackwell andMacQueen (Ann. Statist. 1 (1973) 353-355)) that simplifies implementation andprovide a simulation study for approximating the probabilities of partitionsizes.

A conjugate class of random probability measures based on tilting and with its posterior analysis