Bayesian Prediction in Gamma Models: Admissibility and Infinitesimal Prediction

We study estimation and prediction in the Gamma model $\mathrm{Ga}(\alpha,\beta)$, where the shape parameter $\alpha$ is known and the scale parameter $\beta$ is unknown, under the Kullback--Leibler loss. For $\alpha\le1$, all scale-invariant estimators of $\beta$ have infinite risk, indicating a qualitative change in the estimation problem at the boundary $\alpha=1$. Our main result is that the Bayesian predictive density based on the Jeffreys prior is admissible for all $\alpha>0$. This resolves the admissibility problem for Bayesian predictive densities in Gamma models. As a related result, we also establish the admissibility of the corresponding Bayesian estimator for $\alpha>1$. To prove the predictive admissibility result, we develop an infinitesimal prediction framework based on Gamma processes. This framework naturally leads to a Kullback--Leibler loss for L\'{e}vy densities and establishes a connection between predictive distributions and L\'{e}vy measures. Under the resulting loss, the Bayesian predictive L\'{e}vy density is shown to be the posterior mean L\'{e}vy density. Unlike the normal and Poisson models, infinitesimal prediction in the Gamma model does not reduce to parameter estimation. Instead, it reduces to the estimation of a L\'{e}vy density. We relate this phenomenon to mean mixture curvature and discuss it from an information-geometric viewpoint.