A unified decision model for Bayesian ambiguity and non-affine statistics

This paper studies decision functionals under ambiguity when pure-risk evaluation is determined by a general, possibly non-affine statistic. We represent ambiguity perception by a Bayesian second-order model over first-order models and encode ambiguity attitude with a separate parameter. We formulate four consistency and monotonicity properties for decision functionals combining these objects. We show that second-order averaging does not generally preserve these properties for non-affine statistics, including expected shortfall, the industry-standard risk measure. To address this limitation, we introduce an ambiguity iron, a mapping from a second-order model and an ambiguity attitude to an ambiguity set of first-order models. Worst-case evaluation over this set preserves the chosen statistic as the pure-risk criterion. We construct an ambiguity iron by distorting the second-order model and show that it produces convex and compact sets of predictive models. We derive formulas for several classes of statistics and, in parametric settings, a sample-based approximation for the distorted expectation. Portfolio examples illustrate how ambiguity perception and attitude affect evaluations and allocations. Under posterior concentration and continuity of the parametrization, the ambiguity sets contract, and the associated decision functionals converge to their pure-risk values. We also describe a KL-based robustification for possible model misspecification.