Foundations of neo-Bayesian statistics

We study an axiomatic model of preferences, which contains as special cases Subjective Expected Utility, Choquet Expected Utility, Maxmin and Maxmax Expected Utility and many other models. First, we give a complete characterization of the class of functionals representing these preferences. Then, we show that any such functional can be represented as a Choquet integralI(f)=∫κ(f)dν where κ:B(Σ)→A(C) is the canonical mapping from the space of bounded Σ  -measurable functions into the space of weak∗weak∗-continuous affine functions on a weak*-compact, convex set CC of probability measures on Σ. Conversely, any preference relation defined by means of such functionals satisfies the axioms of the model we study. Different properties of the capacity give rise to different models. Our result shows that the idea of Choquet integration is general enough to embrace all the models mentioned above. In doing so, it widens the range of applicability of well-known procedures in robust statistics theory such as the Neyman–Pearson lemma for capacities [P.J. Huber, V. Strassen, Minimax tests and the Neyman–Pearson lemma for capacities, Ann. Statist. 1 (1973) 251–263], Bayes' theorem for capacities [J.B. Kadane, L. Wasserman, Bayes' theorem for Choquet capacities, Ann. Statist. 18 (1990) 1328–1339] or of results like the Law of Large numbers for capacities [F. Maccheroni, M. Marinacci, A strong law of large numbers for capacities, Ann. Probab. 33 (2005) 1171–1178].