On the iterated a posteriori distribution in Bayesian statistics. (English) Zbl 1150.62334

Recker, F.

Teor. Jmovirn. Mat. Stat. 74, 143-149 (2006).

In Bayesian statistics one has an a priory distribution on the parameter space. After a realization has occurred, one gets the a posteriori distribution. Actually, the realization is often a vector \(x=(x_1,\dots,x_{n})\) and all \(x_{i}\) are assumed to be realizations of i.i.d. random variables. The adequate model is hence to use a product space. The alternative approach is the following. One starts with the above described setting for an experiment. After a sample \(x_1\) has occurred, the a priori distribution is replaced by the respective a posteriori distribution. The second sample \(x_2\) yields again an a posteriori distribution which is taken as the new a priori distribution, and so on. The author calls this concept the iterated a posteriori distribution. It is proved that, under some measurability conditions, both models deliver almost surely the same a posteriori distribution, and this a posteriori distribution does not depend on the ordering of the realizations.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

62F15	Bayesian inference
62C10	Bayesian problems; characterization of Bayes procedures
62C12	Empirical decision procedures; empirical Bayes procedures