Bayes with No Shame: Admissibility Geometries of Predictive Inference
Authors: Nicholas G. Polson, Daniel Zantedeschi
Organizations: Booth School of Business, University of Chicago, Chicago, IL 60637 · Muma College of Business, University of South Florida, Tampa, FL 33620
Modern predictive systems combine predictors, sequential monitors, prediction sets, and online strategies, each with a different certificate of optimality. We study four criterion-relative geometries: Blackwell risk dominance, anytime-valid admissibility, fixed-level marginal coverage with expected-length efficiency within a declared rank-indexed family, and choice-based approachability (CApp) boundary-feasibility. We embed the four procedure types in a common product space and prove witness-based pairwise non-nesting: for every ordered pair of criterion classes, an explicit predictive system is active in both relevant coordinates, belongs to one class, and fails the other. The result records non-nesting across different object spaces and partial orders; it does not assert practical incompatibility. We separate three measure-relative coherence notions. In conditionally i.i.d. models, posterior predictive means under a single prior are martingales under the prior predictive law. For a point null, anytime-valid admissibility within e-processes is equivalent to the nonnegative martingale property. Self-consistency under a predictor's own predictive law does not imply Blackwell admissibility, as shown by a Bernoulli log-loss counterexample. Coverage admissibility is certified by exchangeability ranks within the declared rank-indexed family, while CApp boundary-feasibility uses Cesaro steering. A constrained-Bayes design schema organizes the four paradigms without collapsing their distinct decision spaces, partial orders, or risk functionals. Admissibility is criterion-relative.
Figures & tables
Figure 1: Risk set geometry for ∣Θ∣=2 . The convex risk set R (shaded) maps each decision rule to a risk vector. The lower boundary ∂−R (bold curve) contains all admissible rules. At an admissible point r∗ , the supporting hyperplane (dashed line), with normal π , identifies the prior Π defining the Bayes problem that r∗ solves (Theorem 3.8 ). Interior points are dominated.
Figure 2: Schematic risk set for Bernoulli log-loss prediction with Θ={0.3,0.7} and n=10 . The curve and plotted coordinates are illustrative rather than numerically computed risk values. The boundary points δΠ1 and δΠ2 represent Bayes rules under two full-support priors on this two-point parameter space, depicted by tangent (dashed) supporting hyperplanes. The plug-in MLE p^npi=Sn/n is shown by a finite schematic marker, but its actual extended-real risk vector is (+∞,+∞) : it is dominated because it assigns probability zero to events that occur with positive probability, producing infinite log-loss contributions on {Sn∈{0,n}} . Its extended-real risk coordinates are +∞ at every finite n ; the dot marks a finite proxy position for visual comparison.
Figure 3: Schematic feasible region for anytime-valid inference. An e-process Et starts at E0=1 and satisfies the stopped-value condition of Definition 5.4 ; the shaded region depicts CAV . For a point null, the solid line represents an admissible e-process: a nonnegative martingale (Theorem 5.6 ). The dotted line is a process drawn to violate the stopped-value condition under some P∈H0 , and is therefore infeasible. Stopping at any data-dependent time τ preserves type-I error control at level α via Ville’s inequality: the stopped value Eτ≤1/α with probability at least 1−α (downward arrow). All distinctions in the figure are encoded by line style so that the caption reads correctly in grayscale.
Figure 4: Coverage-feasible region for prediction sets. The horizontal line marks the validity threshold 1−α ; the shaded band above is the feasible class FCov(α) . Among the three points, C lies on the rank-family frontier (conformal and meeting at least the target coverage), O lies below it (oracle Bayes, infeasible), and W lies above it (conservative, feasible but wider than C ).
Geometry
Certificate
Boundary witness
Optimality mode
B : Blackwell
supporting hyperplane
prior Π
pointwise (per-round)
A : Anytime-valid
martingale (point null)
e-process
pathwise (all stopping times)
C : Coverage
exchangeability rank
conformal set
rank-family expected length
D : CApp
Cesàro convergence
regret / redundancy
Cesàro (time-averaged)
Table 1: Summary of the four criteria.
Figure 5: Four admissibility geometries in diamond configuration. Each node represents an admissible class; dashed arrows indicate pairwise non-nesting (Theorems 5.21 and 6.11 ). Blackwell admissibility and CApp boundary-feasibility both involve risk but differ in witness type (prior support vs. Cesàro convergence); anytime-valid and coverage admissibility operate on different procedure spaces entirely.
Procedure
Martingale property
Blackwell admissible
AV- admissible
Coverage admissible
CApp feasible
P1: Bayes shrinkage / KT-type predictor
✓
✓
N/A
N/A
✓
P2: Plug-in MLE Sn/n (unclipped)
✓
×
N/A
N/A
×
P2 ϵ : Clipped plug-in (numerical convention)
׆
not used
N/A
N/A
׆
P3: LR e-process
✓
N/A
✓
N/A
N/A
P4: Conformal prediction set
N/A
N/A
N/A
✓
N/A
Table 2: Bernoulli laboratory: four procedures and four criterion classes. Column headers: Martingale property under the appropriate measure (Definition 4.1 ); Blackwell admissibility ; AV-admissibility (in the sense of Theorem 5.6 ); Coverage admissibility (at level 1−α within the declared rank family, Definition 5.12 ); CApp boundary-feasibility (Definition 6.6 ). ✓ = satisfies; ×= fails; N/A = criterion not applicable to this procedure type. The unclipped plug-in MLE assigns zero probability to realizable events whenever Sn∈{0,n} , giving infinite log loss; its Cesàro-average risk does not converge to H(θ) in the extended-real sense, so it is not CApp boundary-feasible under log loss. † Fixed clipping by ϵ>0 removes the boundary blowup but destroys the plug-in martingale property and is not CApp-feasible on the full model θ∈(0,1) ; it is CApp-feasible only after restricting to θ∈[ϵ,1−ϵ] . We include the clipped row only as a numerical convention (Section 8 ).
n
Bayes risk
MLE clipped ( ϵ=10−6 )
MLE unclipped (theoretical)
Pθ{Sn∈{0,n}}
5
0.694
1.286
+∞
1.70×10−1
10
0.658
0.757
+∞
2.83×10−2
25
0.630
0.633
+∞
1.34×10−4
50
0.621
0.621
+∞
1.80×10−8
100
0.616
0.616
+∞
3.23×10−16
Table 3: Average log loss in nats ( θ=0.3 , B=10,000 ). The unclipped MLE’s log loss is +∞ on every realization where Sn∈{0,n} , an event with strictly positive probability (1−θ)n+θn>0 for every finite n and every θ∈(0,1) ; the theoretical unclipped MLE risk is therefore +∞ uniformly in n . The clipped column applies the convention p^npi↦max(ϵ,min(1−ϵ,p^npi)) with ϵ=10−6 before computing log loss; this is a numerical convention only and should not be interpreted as the true risk. The final column reports the exact theoretical boundary probability Pθ(Sn∈{0,n})=0.7n+0.3n .
Figure 6: The constrained Bayes template in four domains. Each criterion has its own decision space DC , feasible set FC⊆DC , and risk functional RC ; Bayesian integrated risk is optimized within FC (Definition 5.18 ).
This article considers an online version of conformal inference, called adaptive conformal inference [ACI] and introduced by Gibbs and Candès (2021): prediction sets are issued sequentially, after observing features and before the outcomes are revealed. These sets are evaluated both in terms of validity (the fraction of rounds where the outcome was lying in the prediction set) and efficiency (the average lengths of the prediction sets). The two criteria point to different directions (validity favors larger sets). We also target a wide range of scenarios, with exchangeable data and arbitrary data (lack of any stochastic guarantees) as two extremes. A series of existing strategies for ACI typically guarantee that empirical coverage converges to the desired level for arbitrary sequences, but they generally lack simultaneous efficiency guarantees. To provide a unified study, we first formulate ACI as a repeated two-player game with finite action sets and vector-valued payoffs encoding validity and efficiency. Building on this reformulation, we introduce a strategy based on Blackwell approachability and on its opportunistic extension by Bernstein et al. (2014) that ensures validity while adapting the efficiency of the prediction intervals to the underlying degree of stochasticity of the opponent player. The resulting guarantee is "best of many worlds": it recovers the relevant efficiency guarantees in exchangeable and adversarial settings, and provides guarantees in intermediate settings that arise in typical applications such as the forecasting of time series.
Guillaume Principato, Gilles Stoltz
Universit´e Paris-Saclay, CNRS, Inria, Laboratoire de math´ematiques d’Orsay, 91405, Orsay, France · EDF R&D, Palaiseau, France · HEC Paris, Jouy-en-Josas, France
Adaptive conformal inference (ACI) of Gibbs and Cand{è}s and its variants are the standard approach to online conformal prediction under distribution shift, but they suffer from three fundamental limitations. First, their guarantees control only the signed long-run coverage error: persistent miscoverage in one direction can be masked by compensating errors later, so a method can satisfy the theoretical guarantee while being badly wrong for extended periods. Second, existing guarantees say nothing about prediction-set size, so validity can be achieved trivially at the cost of unduly wide prediction sets. Third, the efficiency guarantees that do exist compare against a fixed predictor chosen in hindsight, a benchmark that becomes increasingly less meaningful once the data-generating distribution shifts, since the very notion of an optimal threshold then changes over time. We consider a unified online learning framework that simultaneously controls absolute, non-cancelling coverage violation and prediction-set efficiency against a dynamically evolving benchmark for three important models. In the fully adversarial setting, exploiting the fact that the standard ACI update is exactly projected online gradient descent on the pinball loss, we derive simultaneous coverage and efficiency guarantees for arbitrary monotone Lipschitz efficiency objectives, with no distributional or {\it convexity} assumptions. In the stochastic setting with full-score feedback, we propose a sliding-window quantile tracker and establish a matching minimax lower bound showing our algorithm is rate-optimal. In the covariate-dependent stochastic setting, we develop a partitioned ACI algorithm that tracks a function-valued oracle threshold, and derive simultaneous coverage and efficiency guarantees.
Predictions are increasingly used to guide high-stakes decisions, from treatment selection to policy making. To ensure reliability with imperfect predictions, uncertainty quantification methods such as conformal prediction build prediction sets with coverage guarantees. However, statistical validity alone does not immediately determine the decisions to take, nor the optimality thereof. This gap is especially delicate in counterfactual settings where the outcome that materializes depends on the action taken, so uncertainty cannot be specified independently of the decision rule. We develop a decision-theoretic framework for uncertainty-informed counterfactual decisions. We identify a novel notion of policy-coupled coverage -- namely, coverage of the realized outcome under the action induced by the prediction sets themselves -- as the optimal and lossless interface between uncertainty and action. It plays three roles. First, it justifies acting via a natural max-min rule as minimax-optimal under distributional ambiguity. Second, optimizing prediction sets under policy-coupled coverage is equivalent both to a stronger universal-coverage formulation and to the direct risk-averse optimization over policies and utility certificates; this equivalence yields the explicit form of the population-optimal prediction sets. Third, it admits a two-stage procedure, Policy-Coupled Risk-Averse Conformal Prediction (PC-RACP), that approximates these optimal sets with rigorous finite-sample coverage. Simulations and a real email-marketing experiment confirm that PC-RACP delivers higher utility than existing approaches while maintaining valid coverage, and that ignoring the counterfactual structure of the decision problem is suboptimal for both validity and utility.
Yurui Zheng, Ying Jin
School of Mathematical Sciences, Peking University · Department of Statistics and Data Science, University of Pennsylvania