Fuzzy Conformal Confidence Sets

• Fuzzy conformal confidence sets are a flexible extension of conformal prediction that offer graded exclusion values and probabilistic coverage guarantees.
• They link fuzzy exclusion to e-values, enabling robust hypothesis testing and optimal decision-making under varied risk criteria.
• The framework generalizes error control beyond exchangeable models, supporting post-hoc adaptivity in predictive and statistical tasks.

Fuzzy conformal confidence sets generalize classical conformal prediction by relaxing the strict binary (in/out) structure and enabling a graded measure of exclusion or uncertainty for each prediction. This approach formalizes the intuition that coverage can be interpreted probabilistically or evidentially, making the uncertainty quantification both more flexible and more informative than traditional conformal methods. The concept of the fuzzy conformal set is tied directly to e-values, optimality under minimax (and more general) objectives, and the transfer of error guarantees to subsequent decision-making procedures. This article surveys the mathematical foundations, technical connections, and implications of fuzzy conformal confidence sets as developed in recent research.

1. From Classical to Fuzzy Conformal Confidence Sets

Classical conformal prediction constructs a set-valued mapping from possible predictions to {0,1}, with 0 signifying inclusion (high confidence) and 1 exclusion, such that the miscoverage probability is at most a nominal level α: $\Pr(Z_{n+1} \notin C_\alpha^{Z^n}) \leq \alpha.$ Fuzzy conformal confidence sets extend this by allowing predictions to take values in the interval [0,1], interpreting the output as a degree of exclusion: 0 (full inclusion), 1 (full exclusion), with intermediate values quantifying partial exclusion. The prediction function is thus

$\tilde{C}_\alpha^{Z^n}(z) \in [0, 1].$

The corresponding coverage guarantee is stated in expectation: $$\mathbb{E}[\tilde{C}_\alpha^{Z^n}(Z_{n+1})] \leq \alpha.$$ When the fuzzy set is {0,1}-valued, classical conformal prediction is recovered as a special case. This expectation-based guarantee is a significant generalization, making the set “fuzzy” in its treatment of predictions.

2. E-Values: Linking Fuzziness to Evidence Measures

The meaning of fractional exclusion in fuzzy sets is concretized by linking the degree of exclusion to e-values (evidence values) in hypothesis testing. By rescaling the fuzzy set by 1/α, the fuzzy measure becomes an evidence-valued function: $$\mathcal{E}^{Z^n}(z) = \tilde{C}_\alpha^{Z^n}(z) / \alpha,\qquad \mathcal{E}^{Z^n} : \mathcal{Z} \to [0, 1/\alpha].$$ A value of $1/\alpha$ signifies full exclusion at level α, with $1 / \mathcal{E}^{Z^n}(z)$ giving the smallest α′ at which z is excluded. In this view, $E[\mathcal{E}^{Z^n}(Z_{n+1})] \leq 1$ expresses the standard e-value validity. Theorem 6 in the underlying research formalizes that every valid fuzzy conformal set corresponds to a valid e-value test, and vice versa: $E[\epsilon(Z_{n+1})] \leq 1.$ This duality provides a clean semantics to fuzzy confidence: for each candidate, its degree of exclusion encodes the strength of evidence for excluding it from the set at a given confidence level.

3. Optimality: Testing Perspective and Utility-Based Fuzzy Sets

Classical conformal sets can be interpreted as inverted hypothesis tests. In this perspective, the set-minimization objective—minimizing the expected measure or size of the confidence set—can be cast as an optimal testing problem. Suppose the loss is the expected measure μ(C), then an optimal conformal set $C_\alpha^{Z^n}$ is the Neyman–Pearson most powerful test against an alternative $Q = (P^{Z^n}_*) \otimes \mu^{|Z^n|}$.

For fuzzy sets, optimization proceeds over the e-value space, with more general expected-utility objectives. The Neyman–Pearson utility

$U(x) = \min\{x, 1/\alpha\}$

recovers the classical conformal predictions, but other choices, such as log-utility or power functions, correspond to alternative risk preferences and produce correspondingly different optimal fuzzy sets. The optimal fuzzy conformal set (optimal e-value) for a given utility is characterized (Theorem 7) by: $$\epsilon^U(z^{n+1}) = (U')^{-1}\big(\lambda^{O(z^{n+1})} \cdot LR^{Z_{n+1}|O(z^{n+1})}(z_{n+1})\big),$$ where $LR^{Z_{n+1}|O}$ is the likelihood ratio on orbits (permutation-invariant equivalence classes of the sample), and $\lambda^{O}$ a normalizing constant. The classical construction is recaptured when using the standard Neyman–Pearson utility, while more complex utility functions yield tests adapted to alternative operational criteria.

4. Generalization Beyond Exchangeability

A major result is that fuzzy conformal confidence sets (and their e-value connections) can be constructed under general settings far beyond exchangeable models or classical conformal prediction. For any model family $\mathcal{P}$, validity is achieved so long as the test (e-value) maintains the calibration property uniformly over $\mathcal{P}$: $$E_P[\epsilon(Z_{n+1})] \leq 1 \quad \forall P \in \mathcal{P}.$$ It follows that any valid test or e-value for the true distribution immediately defines a fuzzy prediction confidence set with the same error control. This generalization allows practitioners to recycle hypothesis-testing tools as prediction-set generators for arbitrary (not necessarily i.i.d.) models, making the fuzzy conformal framework widely applicable.

5. Downstream Decision-Making and Post-Hoc Level Adaptivity

Fuzzy prediction sets enable enhanced downstream decisions by supporting both post-hoc level selection and “e-posterior” weighting in loss minimization. For instance:

• In “as-if” decision rules, the decision $\delta(C_\alpha^{Z^n})$ may depend on the set at each α, with guarantees inherited by possibly data-dependent selection of α post-hoc from a family of nested sets.
• In weighted-loss frameworks, decisions may minimize a weighted expected loss

$L(d,z) / \mathcal{E}^{Z^n}(z)$

where the e-value acts as an evidence-dependent weight, downweighting outcomes that strong evidence suggests are implausible. These constructions (which directly generalize minimax and worst-case risk controls for binary sets) extend to fuzzy sets via the e-value calculus, preserving uniform coverage guarantees over all data-dependent level adaptations.

6. Mathematical Formulations and Key Results

Key mathematical results from the theoretical framework include:

• Binary coverage: $\Pr(Z_{n+1} \in C_\alpha^{Z^n}) \geq 1 - \alpha$.
• Fuzzy coverage: $$\mathbb{E}[\tilde{C}_\alpha^{Z^n}(Z_{n+1})] \leq \alpha$$.
• Evidence rescaling: $$\mathcal{E}^{Z^n}(z) = \tilde{C}_\alpha^{Z^n}(z) / \alpha$$, $E[\mathcal{E}^{Z^n}(Z_{n+1})] \leq 1$.
• Sublevel interpretation: for any z, $1 / \mathcal{E}^{Z^n}(z)$ is the “exclusion level” for z.
• Optimality criterion for fuzzy sets as expected-utility maximization under e-value constraints.

These results reveal the mathematical unity behind fuzzy conformal sets, e-value hypothesis testing, and optimal predictive uncertainty quantification.

7. Implications and Significance

Fuzzy conformal confidence sets provide a powerful and principled extension of conformal prediction, with important conceptual and practical implications. They yield a continuum of confidence levels, formalize the linkage between evidence strength and prediction exclusion, support post-hoc and adaptive decisions without loss of error control, and provide a natural interface to existing statistical hypothesis tests via e-values. Their validity holds in both exchangeable and non-exchangeable models, and the approach admits flexible optimization schemes to align with diverse scientific or operational objectives.

This generalized framework synthesizes frequentist prediction with evidence-based and decision-theoretic perspectives, enhancing the expressiveness and applicability of uncertainty quantification in statistical and machine learning problems (Koning et al., 16 Sep 2025).