Large deviations estimates for some level crossing probabilities in Bayesian setting

The probabilities of rare events have interest in some network and risk problems and, in this paper, we concentrate our attention on some level crossing probabilities for Brownian motion and for Poisson process. We are interested in large deviation estimates and we present the Bayesian and the frequentist approaches which lead to different conclusions; more precisely, as argued in other works in the literature on this topic, the Bayesian approach is more conservative. In the Bayesian framework we always deal with conjugate families of prior distributions.