  Suppose that P_{\theta}(g) is a linear functional of a Dirichlet process with
shape \theta H, where \theta >0 is the total mass and H is a fixed probability
measure. This paper describes how one can use the well-known Bayesian prior to
posterior analysis of the Dirichlet process, and a posterior calculus for Gamma
processes to ascertain properties of linear functionals of Dirichlet processes.
In particular, in conjunction with a Gamma identity, we show easily that a
generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet
process is equivalent to the Laplace functional of a class of, what we define
as, Beta-Gamma processes. This represents a generalization of an identity due
to Cifarelli and Regazzini, which is also known as the Markov-Krein identity
for mean functionals of Dirichlet processes. These results also provide new
explanations and interpretations of results in the literature. The identities
are analogues to quite useful identities for Beta and Gamma random variables.
We give a result which can be used to ascertain specifications on H such that
the Dirichlet functional is Beta distributed. This avoids the need for an
inversion formula for these cases and points to the special nature of the
Dirichlet process, and indeed the functional Beta-Gamma calculus developed in
this paper.
