  This article constructs a class of random probability measures based on
exponentially and polynomially tilting operated on the laws of completely
random measures. The class is proved to be conjugate in that it covers both
prior and posterior random probability measures in the Bayesian sense.
Moreover, the class includes some common and widely used random probability
measures, the normalized completely random measures (James (Poisson process
partition calculus with applications to exchangeable models and Bayesian
nonparametrics (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 Terry
Speed (2003) 1-34 IMS)), in a single construction. We describe an augmented
version of the Blackwell-MacQueen P\'{o}lya urn sampling scheme (Blackwell and
MacQueen (Ann. Statist. 1 (1973) 353-355)) that simplifies implementation and
provide a simulation study for approximating the probabilities of partition
sizes.
