Credal Set Correspondence: Theory and Applications

• Credal set correspondence defines precise mappings between sets of probability measures and various imprecise probability representations, such as interval probabilities and capacities.
• It leverages measure-theoretic, convex-geometric, and algorithmic frameworks to enable robust inference and decision-making under uncertainty.
• The framework supports efficient computational methods in Bayesian networks, deep learning model averaging, and decision analyses using lower and upper probability bounds.

Credal set correspondence refers to formal relationships between sets of probability measures (credal sets) and other mathematical objects or representations—most notably, interval-valued probabilities, capacities, random set representations, convex analysis duals, or other imprecise-probability structures. Such correspondences enable transformation, extension, and analysis of imprecise probabilistic reasoning in a mathematically principled way. This article presents the key definitions, constructions, theoretical characterizations, and algorithmic consequences of credal set correspondence, as documented across measure-theoretic, convex-geometric, game-theoretic, and computational frameworks in recent research.

1. Credal Sets: Definitions and Foundational Notions

A credal set is any nonempty, convex, and typically closed (or compact) set of probability measures on a measurable space $(X, \Sigma)$ or, in the discrete case, on a finite set of outcomes or classes. Let $\mathcal{P}(X)$ denote the space of all probability measures on $(X, \Sigma)$. A credal set is then $K \subseteq \mathcal{P}(X)$, compact and convex in the weak topology (Edalat et al., 10 Apr 2026).

For a finite outcome space $Y = \{y_1, \dots y_K\}$, a credal set $\mathcal{K}$ can be described as

$$\mathcal{K} = \bigg\{ p \in \Delta^K : \underline{P}(y_k) \leq p_k \leq \overline{P}(y_k); \sum_{k=1}^K p_k = 1 \bigg\}$$

where $\Delta^K$ is the $K$-simplex, and $[\underline{P}(y_k), \overline{P}(y_k)]$ are lower/upper probability bounds per outcome (Wang et al., 2024).

These sets support robust representations of epistemic uncertainty, encompassing not only precise Bayesian models $\mathcal{P}(X)$0 a singleton$\mathcal{P}(X)$1 but full imprecise-probabilistic modeling as well.

2. Credal Set to Interval Domain Correspondence

A central correspondence is between credal sets and interval-valued probabilities (envelopes) for each event. Given a credal set $\mathcal{P}(X)$2, for each measurable event $\mathcal{P}(X)$3, define the lower and upper probabilities: $\mathcal{P}(X)$4 This maps $\mathcal{P}(X)$5 to the interval $\mathcal{P}(X)$6.

Formally, for the space of credal sets $\mathcal{P}(X)$7, define the envelope map $\mathcal{P}(X)$8: $\mathcal{P}(X)$9 The mapping $(X, \Sigma)$0 is Scott-continuous under the reverse inclusion order on the interval domain $(X, \Sigma)$1 (Edalat et al., 10 Apr 2026).

This mapping extends to capacities (monotone set functions), producing lower and upper capacities that are monotone on $(X, \Sigma)$2, and links to the theory of Choquet integration: $(X, \Sigma)$3 for all non-negative measurable $(X, \Sigma)$4, with analogous results for $(X, \Sigma)$5. Thus, the credal set–interval domain correspondence provides a complete and robust realisation of classical capacity theory and Choquet expected value calculations.

3. Convex, Dual, and Geometric Correspondences

Credal sets admit a variety of geometric and dual representations. In finite dimensions, the duality between credal sets and convex cones of almost desirable gambles $(X, \Sigma)$6 (closed, convex cones in $(X, \Sigma)$7 containing the nonnegative orthant) is given by polarity (Benavoli et al., 2017): $(X, \Sigma)$8 and the intersection $(X, \Sigma)$9 (the probability simplex) yields a credal set. Conversely, any credal set $K \subseteq \mathcal{P}(X)$0 yields the polar cone

$K \subseteq \mathcal{P}(X)$1

This polarity correspondence is a bijection, preserving conditioning and marginalization.

Extensions to lexicographic probabilities (full-rank stochastic matrices representing layers of preference) enable stricter, order-sensitive correspondences for strict desirability cones via lexicographic polarity (Benavoli et al., 2017): $K \subseteq \mathcal{P}(X)$2 with the “$K \subseteq \mathcal{P}(X)$3-credal set” $K \subseteq \mathcal{P}(X)$4 (full-rank stochastic matrices), and their mutual bijection with strict desirability cones.

4. Algorithmic and Computational Correspondences

Credal set correspondences support efficient algorithms for probabilistic inference under imprecise information:

• Credal Sentential Decision Diagrams (CSDDs): Replace precise local probabilities by local credal sets $K \subseteq \mathcal{P}(X)$5 at disjunctive nodes, inducing global credal sets as convex hulls of PSDDs consistent with these constraints. Inference is performed by bottom-up traversal with local LPs, yielding lower and upper marginal (and conditional) probabilities as extrema over the global credal set (Mattei et al., 2020).
• Decision-making under ambiguous evidence: For Bayesian networks, uncertain evidence is encoded as credal virtual or soft evidence; network updating proceeds by augmenting with auxiliary credal nodes, and the computed posterior marginals exactly correspond to the lower and upper bounds obtained via all consistent Bayesian updates (Marchetti et al., 2018).
• Model averaging in deep learning: The credal wrapper transforms $K \subseteq \mathcal{P}(X)$6 predictive distributions from Bayesian neural networks or deep ensembles into a polytope credal set $K \subseteq \mathcal{P}(X)$7 and yields a unique prediction via the intersection-probability transform, which exactly selects the distribution in $K \subseteq \mathcal{P}(X)$8 that is an affine combination of the lower and upper bounds with a data-dependent normalization coefficient (Wang et al., 2024).
• Learning from sets: In credal learning theory, generalization and adaptation bounds for set-valued uncertainties are expressed as the supremum over all $P \