**Sankhya:
The Indian Journal of Statistics**

2003, Volume 65, Pt. 1, 180--196

**Bayesian Calculus For Gamma
Processes With Applications To Semiparametric Intensity Models**

By

Lancelot F. James, Hong Kong University Of Science And Technology

SUMMARY. Explicit calculus for the posterior distribution of convolution mixtures of weighted gamma processes on Polish spaces are derived. This serves to extend the results of Lo and Weng (1989) to a semiparametric setting on arbitrary spaces. The result of this study is applied to two different types of general semiparametric multiplicative intensity models. One in which a prior is constructed based on q conditionally independent weighted gamma measures given a Euclidean parameter and a second dependent model where different hazard rates are based on a common mixing measure. The latter model seems natural for some types of deconvolution models or regression models. As an example, it is shown how this provides a full (implementable) posterior analysis of the Cox regression model. The results also provide the explicit posterior distribution for the Poisson/Gamma random field model considered by Wolpert and Ickstadt (1998a).

*AMS (1991) subject classification. *Primary 62G05; secondary 62F15.

*Key words and phrases. *Convolution mixtures of hazards, weighted
Chinese restaurant process, inhomogeneous Poisson process, multiplicative
intensity models, weighted gamma processes, dependent mixtures of weighted
gamma processes.