🤖 AI Summary
This study addresses the challenge that posterior approximations often fail to preserve downstream decision-making performance by proposing a decision sufficiency framework grounded in Kullback-Leibler divergence contraction. Methodologically, local decision directions are ranked via a generalized eigenvalue problem, thereby decoupling decision coverage from information efficiency. Furthermore, the geometric relationship between the objective loss and baseline approximations is analytically characterized through quadratic local limits on Bayesian action fibers, leveraging Hessian and Fisher information metrics. Ultimately, this work establishes decision-relative criteria for evaluating and designing families of posterior approximations, enabling precise quantification of the decision sufficiency distance.
📝 Abstract
We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior $P$ and loss determine a regret geometry on actions, a baseline approximation $Q_0$ determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family $\mathcal{Q}$ determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian $G$ and a baseline information metric $J_I$ . Their generalized eigenproblem $Gv = γJ_Iv$ orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.