Some Models of Inference in the Risk Theory from a Bayesian Viewpoint

Our purpose is to introduce some models of inference for risk processes. The bayesian viewpoint is adopted and for our treatment the concepts of exchangeability and partial exchangeability (due to B. de Finetti, [6], [7]) are essential.We recall the definitions:The random variables of a sequence (X1, X2 …) are exchangeable if, for every n, the joint distribution of n r.v. of the sequence is always the same, whatever the n r.v. are and however they are permuted.From a structural point of view an exchangeable process X1, X2 … can be intended as a sequence of r.v. equally distributed among which a “stochastic dependence due to uncertainty†exists. More precisely the Xi are independent conditionally on any of a given set (finite or not) of exhaustive and exclusive hypothesis. These hypotheses may concern, for instance, the values of a parameter (number or vector) on which the common distribution, of known functional form, of Xi depends. We shall restrict ourselves to this case. Therefore, we shall assume that, conditionally on each possible value θ of a parameter Θ, the Xi are independent with F(x/θ) as known distribution function. According to the bayesian approach, a probability distribution on Θ must be assigned.