Decision-Sufficient Posterior Approximation

The gist: This paper develops a decision-relative theory of posterior approximation by contracting KL divergence over Bayes-action fibers to produce exact distances to decision adequacy and failure, organizing classical constructions around a specified downstream decision problem

Decision Adequacy and Regret Filtration

The target posterior P and loss determine the decision consequence, defining the regret geometry on actions The Bayes-regret filtration Bϵ(P) is defined as the set of actions where the centered risk ∆P (a) is less than or equal to epsilon, with B0(P) being the Bayes correspondence The zero level B0(P) records exact optimality, while positive levels retain the shape of the loss around the optimum Decision adequacy is defined as an approximate posterior Q being ϵ-adequate if B0(Q) intersects Bϵ(P)

Posterior Accessibility and Information Cost

Posterior accessibility adds a baseline-relative cost by defining the contracted information IQ0(a) as the infimum of forward-KL information required to move from a baseline approximation Q0 to any posterior that makes an action Bayes-optimal The distance to adequacy is defined as RP,Q0(ϵ) = inf IQ0(a) for a in Bϵ(P) The distance to failure is defined by the decision robustness radius MP,Q0(ϵ) = inf IQ0(a) for actions where ∆P (a) > ϵ

Local Decision–Information Geometry

In regular finite-dimensional problems, the target side contributes a regret Hessian G, and the baseline side contributes an information metric JI The generalized eigenproblem Gvi = γiJIvi 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

Restricted Family Diagnostics

For a specified family Q ⊆ CQ0, the diagnostics are defined by two quantities: Coverage asks which first-order decision displacements Q can realize, and Information efficiency asks whether those reachable displacements can be realized at the ambient minimum Fisher cost The quantity ηT measures the worst-case residual regret caused by unavailable action directions over an ambient information ball of radius ρ, and κT is the worst multiplicative Fisher-energy penalty along reachable directions

Oracle Representation and Effective Dimension

The oracle problem in Section 5 identifies which decision directions should be retained when any action-space subspace is available, leading to the effective dimension deff(ϵ; ρ) ϵ

Goal-Oriented Reduction and Examples

The goal-oriented linear–Gaussian reduction of Spantini et al. [11] is recovered as a special case of the decision–information eigenproblem in Section 5 The examples illustrate why this decision-relative view can change a family comparison, showing that the same centered mean-field family can have a pure coverage defect for one decision and complete coverage with excess Fisher cost for another

Conclusion and Extensions

The paper concludes that posterior approximation can be assessed through three coupled objects: the regret induced by action error, the information required to induce that error from a baseline approximation, and the ability of a chosen approximation family to realize the corresponding decision directions The geometry suggests a direct family-design problem where one can seek augmentation to reduce ηT or increase efficiency κT

Regularity Conditions

The quadratic theory in Sections 4–6 uses additional regularity for three distinct purposes: identifying Bayes-action fibers by finite-dimensional equations, differentiating the Bayes-action map in Fisher score space, and localizing the moving constrained optimization problems The exact contracted information value IQ0(a) is given by t 2/2 h TJIh + o(t 2) under certain conditions The local repair and robustness theorem proves that for fixed c ≥ 0, the limiting minimizer is unique, providing exact global accessibility measures The ambient I-projection is an information-theoretic benchmark rather than a claim that every minimizing distribution can arise from the original prior–likelihood model The paper is supported by extensive mathematical machinery connecting classical tools like I-projection, semiparametric efficiency, and approximation theory to posterior approximation

This decision-relative organization yields exact global accessibility measures, local decision–information modes, a tolerance-dependent effective dimension, and separate diagnostics for family coverage and information efficiency. The paper is supported by extensive mathematical machinery connecting classical tools like I-projection, semiparametric efficiency, and approximation theory to posterior approximation

How it works

The construction separates three roles: the target posterior P and loss determine decision consequence, a baseline approximation Q0 determines information accessibility, and a specified approximation family Q determines family representability In regular finite-dimensional problems, these levels reduce locally to a simple picture where the generalized eigenproblem Gvi = γiJIvi ranks action directions by regret consequence per unit information cost

Decision Adequacy and Regret Filtration

The target side contributes a regret Hessian G, while the baseline side contributes an information metric JI The centered risk ∆P (a) records the decision consequence of an action error under the target posterior The regret filtration Bϵ(P) is defined as the set of actions where ∆P (a) is less than or equal to epsilon, with B0(P) being the Bayes correspondence

Posterior Accessibility and Information Cost

The action fibers Ca are defined by the condition a ∈ B0(Q), and IQ0(a) is the contracted information of a relative to Q0, representing the least forward-KL change needed to move from the baseline to any posterior that makes an action Bayes-optimal The distance to adequacy RP,Q0(ϵ) is defined as inf IQ0(a) for a in Bϵ(P)

Local Decision–Information Geometry

The pair (G, JI) determines decision–information modes and an effective dimension The Rayleigh quotient v T Gv / v T JIv shows that γi measures regret consequence per unit information<ref:2610.

1. Proper predictive scoring allows AI systems to maintain identical Bayes-action maps while assigning different regret consequences, enabling task-specific optimization of action selection based on downstream utility or loss structure. This allows for two proper scores can have the same Bayes-action map and the same accessibility metric while assigning different consequences to predictive errors.

2. The framework provides decision-relative criteria for comparing and designing posterior approximation families by separating decision coverage from information efficiency, allowing architects to choose between families that prioritize realizing available moves or those that minimize Fisher cost for reachable displacements.

3. The oracle effective dimension, defined as the smallest rank that guarantees omitted regret at most epsilon, allows systems to dynamically determine the minimum necessary complexity (rank) of an action-space subspace required to support a uniform decision guarantee under specified regret and information scales.

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 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.