Anytime-Valid Admissibility

Updated 4 July 2026

Anytime-valid admissibility is a framework where sequential inference procedures maintain error guarantees at every data-dependent stopping time through nonnegative martingales.
It defines an ambient space of e-processes (the cone C_AV) and uses stopped expectations to compare procedures, ensuring optimality via martingale coherence.
The approach extends to adaptive testing, deadline-sensitive objectives, and filtration-aware evidence combination, highlighting its practical versatility in sequential analysis.

Anytime-valid admissibility is the optimality notion for sequential procedures whose error guarantees must remain valid at every data-dependent stopping time. In the formulation developed in "Bayes with No Shame: Admissibility Geometries of Predictive Inference," its ambient space is the cone $C_{\mathrm{AV}}$ of nonnegative supermartingales under the null, its order is induced by stopped expectations, and its certificate of optimality is martingale coherence: within $C_{\mathrm{AV}}$ , the admissible frontier is characterized exactly by nonnegative martingales under every null distribution (Polson et al., 5 Mar 2026). This martingale-based characterization continues a broader line of work showing that admissible anytime-valid sequential inference must rely on nonnegative martingales, whether expressed as e-processes, p-processes, sequential tests, or confidence sequences (Ramdas et al., 2020).

1. Formal setting and geometric structure

Fix a filtered probability space with filtration $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ , $t\ge 0$ , and a composite null model $H_0$ . The ambient space for anytime-valid admissibility is the cone $C_{\mathrm{AV}}$ of e-processes, defined as processes $E=(E_t)_{t\ge 0}$ with $E_0=1$ such that, under every $P\in H_0$ , $E_t\ge 0$ for all $C_{\mathrm{AV}}$ 0 and

$C_{\mathrm{AV}}$ 1

for all $C_{\mathrm{AV}}$ 2. Equivalently,

$C_{\mathrm{AV}}$ 3

This equivalence makes the feasibility constraint explicitly stopping-time robust (Polson et al., 5 Mar 2026).

The operational consequence is Ville-type control: $C_{\mathrm{AV}}$ 4 for each $C_{\mathrm{AV}}$ 5. Hence every $C_{\mathrm{AV}}$ 6 yields an anytime-valid level- $C_{\mathrm{AV}}$ 7 test by rejecting when $C_{\mathrm{AV}}$ 8. In this geometry, admissibility is not defined by a convex risk frontier but by a cone of feasible capital paths whose defining constraint is optional-stopping validity (Polson et al., 5 Mar 2026).

The partial order is likewise sequential. Within $C_{\mathrm{AV}}$ 9, a procedure $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 0 dominates $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 1 if

$\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 2

for all $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 3 and all stopping times $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 4, with strict inequality for some $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 5. This order compares procedures through their stopped expectations and corresponding rejection behavior under the same time-uniform Type I constraint. A plausible implication is that the relevant notion of “better” is intrinsically pathwise and stopping-time indexed, not reducible to a single fixed-horizon power number.

An e-process is any element of $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 6. In the game-theoretic statistics literature, e-values are nonnegative random variables with null expectation at most $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 7; e-processes are their sequential analogues, preserving the same guarantee under optional stopping. A prototypical construction is the predictable likelihood-ratio capital process

$\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 8

where $\mathcal{F}_t=\sigma(X_1,\dots,X_t)$ 9 is a null density and each $t\ge 0$ 0 is predictable, that is, $t\ge 0$ 1-measurable. Under any $t\ge 0$ 2,

$t\ge 0$ 3

so $t\ge 0$ 4 is a nonnegative martingale. Test inversion applied to a family $t\ge 0$ 5 yields confidence sequences

$t\ge 0$ 6

which satisfy time-uniform coverage (Polson et al., 5 Mar 2026).

The 2020 martingale-universality results place e-processes in a larger hierarchy of anytime-valid instruments. A $t\ge 0$ 7-e-process satisfies $t\ge 0$ 8 for all stopping times $t\ge 0$ 9; a $H_0$ 0-p-process satisfies $H_0$ 1; sequential tests and confidence sequences can be converted into one another by thresholding and inversion. In particular, if $H_0$ 2 is an e-process, then $H_0$ 3 is a p-process, and admissible confidence sequences can be obtained by inverting admissible martingale threshold tests (Ramdas et al., 2020).

A distinct but related route to anytime validity is sequentialization of a fixed- $H_0$ 4 test. Given a simple null and a valid terminal test $H_0$ 5, the process

$H_0$ 6

is a nonnegative martingale and hence anytime-valid, while matching the original test at time $H_0$ 7 whenever $H_0$ 8 is $H_0$ 9-measurable (Koning et al., 7 Jan 2025). This shows that Doob-style martingale constructions are not merely one design pattern among many; they also recover conventional fixed-horizon procedures in sequential form.

Concrete admissible constructions inside $C_{\mathrm{AV}}$ 0 include predictable likelihood-ratio martingales, mixture e-processes

$C_{\mathrm{AV}}$ 1

and betting-style capital processes of the form

$C_{\mathrm{AV}}$ 2

where $C_{\mathrm{AV}}$ 3 is a conditional e-factor with $C_{\mathrm{AV}}$ 4 under $C_{\mathrm{AV}}$ 5. Convex mixtures preserve feasibility, but unless conditional-expectation equality is retained they typically yield supermartingales rather than martingales, suggesting a potential loss of admissibility (Polson et al., 5 Mar 2026).

3. Admissibility certificate: martingale coherence

The central theorem of the AV geometry is exact: within $C_{\mathrm{AV}}$ 6, a procedure is admissible if and only if it is a nonnegative martingale under every $C_{\mathrm{AV}}$ 7. In the terminology of the 2026 geometry paper, martingale coherence is therefore necessary and sufficient for anytime-valid admissibility within e-processes (Polson et al., 5 Mar 2026).

The intuitive structure is asymmetric. If $C_{\mathrm{AV}}$ 8 is a strict supermartingale, meaning that

$C_{\mathrm{AV}}$ 9

with positive probability for some $E=(E_t)_{t\ge 0}$ 0 and some $E=(E_t)_{t\ge 0}$ 1, then there exists another feasible process in $E=(E_t)_{t\ge 0}$ 2 that improves the stopped expectations uniformly without violating the validity constraint; such a process is dominated. Conversely, if $E=(E_t)_{t\ge 0}$ 3 is a nonnegative martingale under every null, any attempt to raise its stopped expectations under $E=(E_t)_{t\ge 0}$ 4 would violate the supermartingale feasibility condition. The martingale property is thus not merely sufficient evidence of validity; it is the exact witness of frontier membership in the cone.

The earlier universality theorem is sharper in historical perspective. For point nulls, admissible e-processes are exactly nonnegative martingales with unit mean; admissible sequential tests are threshold tests on running suprema of martingales with no overshoot; admissible p-processes are closed max-martingales with a uniform limit law; and admissible confidence sequences arise by inverting admissible martingale tests (Ramdas et al., 2020). This broader framework shows why nonnegative supermartingales are ubiquitous in safe sequential inference: supermartingales certify feasibility, but admissibility forces the stronger martingale boundary condition.

A frequent misconception is that martingale coherence is a universal optimality criterion across all forms of predictive inference. The geometry results reject that view. Martingale coherence is necessary and sufficient for anytime-valid admissibility within $E=(E_t)_{t\ge 0}$ 5, necessary but not sufficient for Blackwell admissibility, and not necessary for marginal coverage validity or Cesàro approachability admissibility (Polson et al., 5 Mar 2026). The same martingale can therefore be frontier-optimal in one criterion and irrelevant, or even insufficient, in another.

4. Criterion-relative admissibility and geometric separation

The 2026 geometry paper places anytime-valid admissibility beside three other admissibility notions: Blackwell risk dominance, marginal coverage validity, and Cesàro approachability admissibility. All four fit a common constrained-Bayes template—minimize Bayesian risk subject to feasibility—but the feasible sets, orders, and certificates live in different spaces, making the resulting frontiers geometrically incompatible (Polson et al., 5 Mar 2026).

Geometry	Space and order	Certificate
Blackwell	Convex risk set; coordinatewise dominance	Supporting-hyperplane prior
Anytime-valid	Cone of nonnegative supermartingales; stopped-expectation/type-I order	Nonnegative martingale
Coverage	Exchangeable prediction sets; coverage-level feasibility	Exchangeability rank
CAA	Time-averaged risk set; Cesàro approach to the boundary	Approachability/steering argument

The criterion-separation theorem states that the admissible classes are pairwise non-nested. In the notation of the paper, $E=(E_t)_{t\ge 0}$ 6, $E=(E_t)_{t\ge 0}$ 7, and $E=(E_t)_{t\ge 0}$ 8 are pairwise non-nested, and the extended theorem adds $E=(E_t)_{t\ge 0}$ 9 for CAA-admissibility, with the four classes again pairwise non-nested. The examples are concrete: a Bayes point predictive rule can be Blackwell admissible while lying outside $E_0=1$ 0; a likelihood-ratio e-process can be AV-admissible while lying outside $E_0=1$ 1; conformal prediction sets achieve marginal coverage while lying outside both $E_0=1$ 2 and $E_0=1$ 3; and defensive forecasting is CAA-admissible while being neither Bayes per round, nor an e-process, nor a prediction set (Polson et al., 5 Mar 2026).

This criterion-relativity has substantive consequences. No common refinement of the partial orders exists across these geometries, because the objects being ordered differ: point predictions, supermartingale capital paths, prediction sets, and time averages. A plausible implication is that disagreements over “optimality” in sequential inference often reflect mismatched feasibility classes rather than contradictory theorems. Under this view, admissibility is not a universal badge but a property indexed by the inferential task and the validity constraint.

Several recent developments preserve the anytime-valid core while changing the objective relative to which admissibility is assessed. In "Time-sensitive anytime-valid testing," the objective is not eventual rejection alone but

$E_0=1$ 4

where $E_0=1$ 5 is a non-increasing reward on rejection times. For hard deadlines $E_0=1$ 6, the simple-vs-simple problem reduces to a finite-horizon Neyman–Pearson event, and the optimal anytime-valid procedure is the associated Doob e-process. For exponentially decaying rewards, the stationary approximation yields the exponential-decay-optimal criterion, or EDO, which is first-order optimal in $E_0=1$ 7 and converges to the classical growth-rate-optimal viewpoint as $E_0=1$ 8 (Clerico et al., 7 May 2026). Here admissibility remains anytime-valid but becomes explicitly reward-relative.

A different objective appears in "Towards Anytime-Valid Statistical Watermarking." There the detector chooses a valid e-value $E_0=1$ 9 against a composite null defined by independence and an $P\in H_0$ 0-ball around an anchor distribution $P\in H_0$ 1, while optimizing worst-case expected log-growth: $P\in H_0$ 2 The paper derives the optimal anchored e-value

$P\in H_0$ 3

proves the optimal worst-case log-growth rate $P\in H_0$ 4, and shows that the asymptotic sample complexity satisfies $P\in H_0$ 5 while $P\in H_0$ 6 for any valid $P\in H_0$ 7 (Huang et al., 19 Feb 2026). This is an admissibility statement under a sequential growth/sample-efficiency criterion rather than the stopped-expectation order of $P\in H_0$ 8.

Anytime-validity also interacts nontrivially with filtration choice. "Combining Evidence Across Filtrations" shows that an e-process valid in a coarser filtration need not remain valid in a finer one, so naive averaging across filtrations can fail. The remedy is an adjuster $P\in H_0$ 9 applied to the running maximum: $E_t\ge 0$ 0 which lifts a coarse-filtration e-process into a finer filtration; adjust-then-combine procedures then recover a valid e-process in the finest filtration of interest (Choe et al., 2024). The paper also proves a characterization theorem for adjusters and identifies a logarithmic cost to recovering validity in the original filtration. This suggests that admissible anytime-valid evidence combination depends not only on null models and stopping rules but also on the information structure relative to which stopping is defined.

6. Applications, limitations, and open directions

The most direct contemporary applications of anytime-valid admissibility are in adaptive AI systems, where optional stopping is endogenous rather than externally imposed. "PACE: Anytime-Valid Acceptance Tests for Self-Evolving Agents" recasts each commit decision as a paired sequential hypothesis test. Using discordant paired outcomes $E_t\ge 0$ 1, it defines

$E_t\ge 0$ 2

with default $E_t\ge 0$ 3, and commits when $E_t\ge 0$ 4. Under the null condition

$E_t\ge 0$ 5

the process is a nonnegative supermartingale, so

$E_t\ge 0$ 6

The paper is explicit that it does not claim run-level FWER/FDR control, uniformly most powerful status, or formal admissibility among all anytime-valid tests; the contribution is per-candidate false-commit control under optional stopping, together with early stopping and lower evaluation cost (Shawn, 6 Jun 2026). This is a useful practical distinction: validity may be exact while optimality remains criterion- and class-dependent.

"Self-Evolving Agents with Anytime-Valid Certificates" embeds anytime-valid gates into a multi-layer architecture for self-modification. Harness edits are accepted through a paired-difference confidence sequence with performativity correction; reward-model updates use a Hoeffding e-process; continual adapter updates require a time-uniform PAC-Bayes forgetting bound and a performative trust-region condition. Confirmations spend slices of a global error budget through the normalized horizon-free confirm-triggered harmonic schedule

$E_t\ge 0$ 7

The system emits structured certificates per round, but the paper also emphasizes that the composition of these guarantees under endogenous proposal and performative shifts is not itself proved (Sengupta, 1 Jul 2026). This marks an important boundary: anytime-valid admissibility at the component level does not automatically imply a full-system theorem under nested adaptivity.

Across the literature, several limitations recur. Anytime-valid admissibility depends on correct specification of the null $E_t\ge 0$ 8, on filtration measurability, and on the stopping-time model under which optional-stopping validity is asserted (Polson et al., 5 Mar 2026). Aggregation is delicate because convex mixtures preserve feasibility but can move a martingale frontier point into the interior of the supermartingale cone (Polson et al., 5 Mar 2026). Time-sensitive testing leaves open robust criteria for composite alternatives and multi-threshold objectives (Clerico et al., 7 May 2026). Anchored e-watermarking depends on anchor calibration and null independence assumptions (Huang et al., 19 Feb 2026). Cross-filtration lifting incurs an unavoidable logarithmic insurance cost (Choe et al., 2024).

The common synthesis is that anytime-valid admissibility is not a synonym for generic sequential validity. It refers to frontier-optimality inside a feasibility class defined by optional-stopping-safe evidence processes. In its canonical form, that frontier is the set of nonnegative martingales under the null (Polson et al., 5 Mar 2026). Beyond that canonical form, the same anytime-valid discipline supports deadline-sensitive objectives, worst-case log-growth criteria, filtration-aware evidence combination, and adaptive decision gates, but the admissibility claim is always indexed by the objective, the feasible class, and the underlying information structure.