AI 中文总结
该研究提出决策充分的后验近似框架，通过广义特征问题等方法，为后验近似族的比较与设计提供决策相关准则。
AI 中文摘要
我们研究要求后验近似保留指定下游决策问题的后果。目标后验P和损失决定了动作上的遗憾几何，基线近似Q0决定了诱导动作变化所需的前向Kullback-Leibler信息，受限近似族Q决定了可用的此类变化。在贝叶斯动作纤维上收缩KL散度，可得到到决策充分性和决策失败的精确距离，以及到达决策边界任一侧的信息最少的后验变形。在正则有限维问题中，目标和基线构造具有二次局部极限：目标遗憾海森矩阵G和基线信息度量JI。它们的广义特征问题Gv=γJIv按每单位信息成本的遗憾后果对局部决策方向排序，并诱导出依赖于容差的有效维度。对于受限近似族，切图像将决策覆盖与信息效率分离开：一个族可能会错过重要的决策方向，或者仅以超额Fisher成本实现可达方向。所得框架为比较和设计后验近似族提供了与决策相关的准则。
英文摘要
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.
Comments45 pages, 1 figure
