E-admissibility: Robust Criterion in Decision Theory

• E-admissibility is a decision criterion that identifies undominated actions by requiring an option to maximize expected utility for at least one probability distribution in a credal set.
• In merging e-values, only weighted arithmetic means meet the E-admissibility standard, ensuring valid evidence aggregation in anytime-valid inference and multiple testing.
• ε-admissibility extends this framework by quantifying the robustness of estimators, offering a second-order refinement beyond classical minimax criteria in high-dimensional settings.

E-admissibility is a foundational concept in statistical decision theory and the theory of hypothesis testing, with central roles in robust choice under uncertainty, the merging of evidential quantities, and the formal refinement of admissibility and minimaxity in high-dimensional statistics. It arises in multiple guises: as a selection criterion for actions (or estimators) under ambiguous beliefs, as the core admissibility notion for combining e-values, and as an analytical tool for comparing procedures beyond leading-order minimax risk.

1. Definitions and Basic Principles

Let $\Omega$ be a finite outcome space, $S$ a finite set of acts (or estimators), and $u: S \times \Omega \to \mathbb{R}$ a utility function. Given a credal set $P$ (a nonempty, convex set of probability mass functions on $\Omega$), an act $x \in S$ is E-admissible if there exists at least one $p \in P$ such that $x$ attains the maximum expected utility in $S$ under $p$; that is,

$S$0

This notion, rooted in Levi (1978) and extended in contemporary decision theory, encompasses both complete and incomplete information by characterizing undominated choices given ambiguous beliefs over states.

In the context of e-values—a major tool in recent developments in anytime-valid inference and multiple testing—E-admissibility characterizes undominated merging rules for combining evidence against a null hypothesis. E-values are nonnegative random variables $S$1 with $S$2 for all $S$3 in a hypothesis family $S$4, typically with $S$5 atomless, so $S$6 is simply a $S$7-valued variable with mean at most one. An e-merging function $S$8 is E-admissible if it is not strictly dominated by any other merging function on a set of positive measure; see (Wang, 2024).

A related and increasingly important generalization is $S$9-admissibility in statistical estimation, where an estimator $u: S \times \Omega \to \mathbb{R}$0 is $u: S \times \Omega \to \mathbb{R}$1-admissible if there does not exist any competitor that uniformly improves upon its risk by more than $u: S \times \Omega \to \mathbb{R}$2; see (Yano et al., 2017).

2. Choice Function Framework and Rejection Assessments

To formalize partial or incomplete preference information (e.g., from finite assessments of rejected or accepted options), E-admissibility naturally extends to the setting of choice functions. A choice function $u: S \times \Omega \to \mathbb{R}$3 assigns to each finite, nonempty set $u: S \times \Omega \to \mathbb{R}$4 of options a (possibly strict) subset $u: S \times \Omega \to \mathbb{R}$5 of undominated options. Partial rejection information is encoded as an assessment $u: S \times \Omega \to \mathbb{R}$6, a collection of pairs $u: S \times \Omega \to \mathbb{R}$7 with $u: S \times \Omega \to \mathbb{R}$8, interpreted as "from $u: S \times \Omega \to \mathbb{R}$9, all $P$0 are definitely rejected" (Decadt et al., 2022).

Given such an assessment, the family of compatible credal sets is

$P$1

The unique most conservative E-admissible extension $P$2 rejects no option except as forced by $P$3, and can be computed via a finite family of linear feasibility problems; $P$4 if and only if there exists a $P$5 with $P$6 for all $P$7.

For computational purposes, the feasibility characterization employs sets of utility differences $P$8 and searches for feasible $P$9 satisfying strict inequalities, operationalizing E-admissibility as a tractable decision rule (Decadt et al., 2022).

3. E-admissibility in Merging E-values

In the context of multiple e-values $\Omega$0, admissibility criteria for their combination are central to anytime-valid inference and modern multiple testing. A $\Omega$1-ary e-merging function $\Omega$2 maps $\Omega$3 e-values (respecting any dependency structure) to another valid e-value: $\Omega$4 must itself meet the e-value criterion.

The definitive structure theorem, proven via an application of multi-marginal optimal transport and Sion’s minimax theorem, states that the only admissible e-merging functions are weighted arithmetic means (possibly with an added constant to ensure normalization). Specifically, for the standard simplex in $\Omega$5,

$\Omega$6

one forms

$\Omega$7

with $\Omega$8 admissible if and only if $\Omega$9 for some $x \in S$0. Thus, all admissible merging rules are convex combinations of the input coordinates and the constant function. Nonlinear or rank-based merging rules cannot be E-admissible in this sense. When symmetry is imposed, the only symmetric admissible merging is a convex combination of the arithmetic mean and the constant function (Wang, 2024).

4. ε-Admissibility and Its Distinction From Classical Criteria

In statistical estimation, entirely distinct but closely related is the notion of $x \in S$1-admissibility. Given a statistical model $x \in S$2 and estimator $x \in S$3, the weak admissibility gap is

$x \in S$4

where $x \in S$5 is the risk. An estimator is $x \in S$6-admissible if $x \in S$7. This quantifies, in a uniform sense, how robust an estimator is to uniform improvement, with classical admissibility recovered at $x \in S$8. Minimax-optimal estimators may both be rate-optimal, but their admissibility gaps can differ by orders of magnitude in high dimension or nonparametrics, underscoring that minimaxity alone may be insufficient to distinguish robustness (Yano et al., 2017).

5. Computational Approaches

For choice-function-based E-admissibility, the computational task boils down to linear feasibility: for each candidate action, and each possible selection of pairwise differences consistent with assessments, solve

$x \in S$9

$p \in P$0

$p \in P$1

Feasibility of this system certifies E-admissibility for a specific option; infeasibility certifies rejection. Primal and dual encodings (via Farkas’ lemma) are available for efficient implementation (Decadt et al., 2022).

For ε-admissibility in high-dimensional estimation, closed-form tight lower and upper bounds can be obtained via Bayes-risk comparisons and domination gaps. For example, in the high-dimensional Poisson model and Gaussian sequence settings, explicit calculations demonstrate that certain shrinkage or mixture strategies furnish estimators that are exponentially ε-admissible, while more “classical” Bayes estimators fail this stringent robustness test (Yano et al., 2017).

6. Applications and Implications

E-admissibility for merging e-values provides a rigorous justification for the widespread use of (possibly weighted) arithmetic means in the aggregation of evidential quantities. In situations such as p-value merging (via e-value intermediaries), anytime-valid inference, discovery matrices, and knockoff derandomization, only weighted arithmetic averages achieve E-admissibility, precluding all nonlinear or data-dependent merging approaches if dominance is to be avoided (Wang, 2024).

In decision analysis under ambiguity, E-admissibility remains the principled rule for identifying undominated acts when only partial rejection information is available, ensuring that no potentially optimal act is excluded unless forced by explicit evidence (Decadt et al., 2022).

Analogously, ε-admissibility provides a finer second-order selection criterion among minimax-optimal estimators in high-dimensional and nonparametric contexts, where ties in minimax rates are ubiquitous. The uniform gap characterized by ε-admissibility reveals essential distinctions in the global risk landscape, informing the selection of genuinely robust procedures (Yano et al., 2017).

7. Summary Table: E-admissibility Across Contexts

Domain	E-admissibility Criterion	Main Consequence
Choice under ambiguity	Exists $p \in P$2 s.t.\ $p \in P$3 maximizes expected utility in $p \in P$4	Selects undominated acts; coarsest extension of “hard” rejections
Merging e-values	Weighted arithmetic mean of e-values (plus, possibly, a constant)	No nonlinear merging is E-admissible; justifies arithmetic averaging
$p \in P$5-admissibility	No uniform competitor with risk improvement $p \in P$6 everywhere	Second-order refinement beyond minimax rate; quantifies robustness

E-admissibility thus occupies a central role as both a fundamental property of rational choice under uncertainty and a powerful technical criterion for admissibility and robustness in statistical decision-making, estimation, and evidence aggregation (Wang, 2024, Decadt et al., 2022, Yano et al., 2017).