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This paper is devoted to Bayesian analysis for semi-parametric proportional hazard models (also called Cox models). Independent observations are made, whose distributions are modifications of a common baseline distribution through multiplicative coefficients for the hazard function (called stresses, or constraints) which may vary from one observation to another. A dirichlet prior is considered for the unknown baseline distribution. Under mild derivability conditions on this prior, and for fixed values of the constraints, the predictive distribution (also called the integrated distribution) of observations is a particular case of the so-called multivariate piecewise regular measures. Such measures are not absolutely continuous w.r.t. Lebesgue measure, but we show that their restrictions to subsets characterized by equalities and strict inequalities between the variables are. We give an algorithm and an explicit formula for the computation of the densities of these restrictions. For general piecewise regular measures, the computations using this formula lies on a combinatorial procedure for which we have written a program (with Mathematica). For the predictive distribution in the Cox-Dirichlet model, we give a simplified formula whose use implies only elementary computations.
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