  In a recent paper, the authors studied the distribution properties of a class
of exchangeable processes, called measure-valued P\'{o}lya sequences (MVPS),
which arise as the observation process in a generalized urn sampling scheme.
Here we present several results in the form of "sufficientness" postulates that
characterize their predictive distributions. In particular, we show that
exchangeable MVPSs are the unique exchangeable models whose predictive
distributions are a mixture of the marginal distribution and the average of a
probability kernel evaluated at past observation. When the latter coincides
with the empirical measure, we recover a well-known result for the exchangeable
model with a Dirichlet process prior. In addition, we provide a "pure"
sufficientness postulate for exchangeable MVPSs that does not assume a
particular analytic form for the predictive distributions. Two other
sufficientness postulates consider the case when the state space is finite.
