Posterior Gamma process samples by updating prior samples

\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\cov}{\operatorname{Cov}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\dif}{\backslash} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Can we find a transformation that will turn a Gamma process prior sample into a Gamma process posterior sample?

What are the summary statistics for such a process with respect to which we could construct an update? Would it be in terms of rate changes or addition of other processes?