Prediction rules for exchangeable sequencesrelated to species sampling 1

Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the rst n terms of the sequence, the next term equals the jth distinct value observed so far with probability pj;n, for j = 1; 2; : : :, and otherwise is a new value with distribution for some probability measure on S with no atoms. Then the pj;n depend only on the partitition of the rst n integers induced by the rst n values of the sequence. All possible distributions for such an exchangeable sequence are characterized in terms of constraints on the pj;n and in terms of their de Finetti representations.