A bayesian approach to a reliability problem: Theory,analysis and interesting numerics

We consider approximate Bayesian inference about the quantity R = P[Y₂> Y₁] when both the random variables Y₁, Y₂ have expectations that depend on certain explanatory variables. Our interest centers on certain characteristics of the posterior of R under Jeffreys's prior, such as its mean, variance and percentiles. Since the posterior of R is not available in closed form, several approximation procedures are introduced, and their relative performance is assessed using two real datasets.