---
title: Anytime-Valid Admissibility
url: https://www.emergentmind.com/topics/anytime-valid-admissibility
type: topic
---

# Anytime-Valid Admissibility

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 [2603.05335]. 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 [2009.03167].

## 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 \(t\) and
\[
\mathbb{E}_P[E_t\mid \mathcal{F}_{t-1}] \le E_{t-1}\quad\text{a.s.}
\]
for all \(t\). Equivalently,
\[
C_{\mathrm{AV}}=\{E: E_t\ge 0\ \text{and}\ \sup_{P\in H_0}\mathbb{E}_P[E_\tau]\le 1\ \text{for every stopping time }\tau\}.
\]
This equivalence makes the feasibility constraint explicitly stopping-time robust [2603.05335].

The operational consequence is Ville-type control:
\[
P\!\left(\sup_{t\ge 0} E_t \ge 1/\alpha\right)\le \alpha,\qquad \alpha\in(0,1),
\]
for each \(P\in H_0\). Hence every \(E\in C_{\mathrm{AV}}\) yields an anytime-valid level-\(\alpha\) test by rejecting when \(E_t\ge 1/\alpha\). 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 [2603.05335].

The partial order is likewise sequential. Within \(C_{\mathrm{AV}}\), a procedure \(E\) dominates \(E'\) if
\[
\mathbb{E}_P[E_\tau]\ge \mathbb{E}_P[E'_\tau]
\]
for all \(P\in H_0\) and all stopping times \(\tau\), with strict inequality for some \((P,\tau)\). 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.

## 2. E-processes, related instruments, and canonical constructions

An e-process is any element of \(C_{\mathrm{AV}}\). In the game-theoretic statistics literature, e-values are nonnegative random variables with null expectation at most \(1\); e-processes are their sequential analogues, preserving the same guarantee under optional stopping. A prototypical construction is the predictable likelihood-ratio capital process
\[
E_t=\prod_{s=1}^t L_s,\qquad L_s=\frac{f_{\vartheta_s}(X_s)}{f_0(X_s)},
\]
where \(f_0\) is a null density and each \(\vartheta_s\) is predictable, that is, \(\mathcal{F}_{s-1}\)-measurable. Under any \(P\in H_0\),
\[
\mathbb{E}_P[L_s\mid \mathcal{F}_{s-1}]=1,
\]
so \((E_t)\) is a nonnegative martingale. Test inversion applied to a family \(\theta\mapsto E_t(\theta)\) yields confidence sequences
\[
C_t=\{\theta:\sup_{s\le t}E_s(\theta)<1/\alpha\},
\]
which satisfy time-uniform coverage [2603.05335].

The 2020 martingale-universality results place e-processes in a larger hierarchy of anytime-valid instruments. A \(Q\)-e-process satisfies \(\mathbb{E}_Q[E_\tau]\le 1\) for all stopping times \(\tau\); a \(Q\)-p-process satisfies \(Q(p_\tau\le \alpha)\le \alpha\); sequential tests and confidence sequences can be converted into one another by thresholding and inversion. In particular, if \((E_t)\) is an e-process, then \(p_t:=1\wedge \inf_{s\le t}1/E_s\) is a p-process, and admissible confidence sequences can be obtained by inverting admissible martingale threshold tests [2009.03167].

A distinct but related route to anytime validity is sequentialization of a fixed-\(N\) test. Given a simple null and a valid terminal test \(\varepsilon\), the process
\[
\varepsilon_n:=\mathbb{E}^{\mathbb{P}}[\varepsilon\mid \mathcal{F}_n]
\]
is a nonnegative martingale and hence anytime-valid, while matching the original test at time \(N\) whenever \(\varepsilon\) is \(\mathcal{F}_N\)-measurable [2501.03982]. 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 \(C_{\mathrm{AV}}\) include predictable likelihood-ratio martingales, mixture e-processes
\[
E_t=\int \prod_{s=1}^t \frac{f_\theta(X_s)}{f_0(X_s)}\,d\Lambda(\theta),
\]
and betting-style capital processes of the form
\[
E_t=E_{t-1}\cdot M_t,\qquad E_0=1,
\]
where \(M_t\) is a conditional e-factor with \(\mathbb{E}_P[M_t\mid \mathcal{F}_{t-1}]\le 1\) under \(H_0\). Convex mixtures preserve feasibility, but unless conditional-expectation equality is retained they typically yield supermartingales rather than martingales, suggesting a potential loss of admissibility [2603.05335].

## 3. Admissibility certificate: martingale coherence

The central theorem of the AV geometry is exact: within \(C_{\mathrm{AV}}\), a procedure is admissible if and only if it is a nonnegative martingale under every \(P\in H_0\). In the terminology of the 2026 geometry paper, martingale coherence is therefore necessary and sufficient for anytime-valid admissibility within e-processes [2603.05335].

The intuitive structure is asymmetric. If \(E\) is a strict supermartingale, meaning that
\[
\mathbb{E}_P[E_t\mid \mathcal{F}_{t-1}]<E_{t-1}
\]
with positive probability for some \(t\) and some \(P\in H_0\), then there exists another feasible process in \(C_{\mathrm{AV}}\) that improves the stopped expectations uniformly without violating the validity constraint; such a process is dominated. Conversely, if \(E\) is a nonnegative martingale under every null, any attempt to raise its stopped expectations under \(H_0\) 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 [2009.03167]. 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 \(C_{\mathrm{AV}}\), necessary but not sufficient for Blackwell admissibility, and not necessary for marginal coverage validity or Cesàro approachability admissibility [2603.05335]. 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 [2603.05335].

| 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, \(\mathfrak{B}\), \(\mathfrak{A}\), and \(\mathfrak{C}\) are pairwise non-nested, and the extended theorem adds \(\mathfrak{D}\) 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 \(\mathfrak{A}\); a likelihood-ratio e-process can be AV-admissible while lying outside \(\mathfrak{B}\); conformal prediction sets achieve marginal coverage while lying outside both \(\mathfrak{A}\) and \(\mathfrak{B}\); and defensive forecasting is CAA-admissible while being neither Bayes per round, nor an e-process, nor a prediction set [2603.05335].

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.

## 5. Objective-specific refinements and extensions

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
\[
\mathbb{E}_{\pi_1^{\otimes}}[R(\tau)],
\]
where \(R\) is a non-increasing reward on rejection times. For hard deadlines \(R(t)=\mathbf{1}\{t\le T\}\), 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 \(\alpha\) and converges to the classical growth-rate-optimal viewpoint as \(T\to\infty\) [2605.06521]. 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 \(e\) against a composite null defined by independence and an \(L_1\)-ball around an anchor distribution \(p_0\), while optimizing worst-case expected log-growth:
\[
\sup_{e\in E}\inf_{q\in Q(p_0,\delta)}\sup_{w\in P(p_0,q)} \mathbb{E}_w[\log e(V,S)].
\]
The paper derives the optimal anchored e-value
\[
e^*(v,s)=
\begin{cases}
(1-\delta/2)/p_0(s), & s=v,\\[4pt]
[\delta/(2(n-1))]/p_0(s), & s\ne v,
\end{cases}
\]
proves the optimal worst-case log-growth rate \(J^*\), and shows that the asymptotic sample complexity satisfies \(SC(e^*)=1/J^*\) while \(SC(e)\ge 1/J^*\) for any valid \(e\) [2602.17608]. This is an admissibility statement under a sequential growth/sample-efficiency criterion rather than the stopped-expectation order of \(C_{\mathrm{AV}}\).

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 \(A\) applied to the running maximum:
\[
e_t^{\mathrm{adj}}=A\!\left(\sup_{i\le t} e_i\right),
\]
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 [2402.09698]. 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 \(w_i\in\{0,1\}\), it defines
\[
E_i=E_{i-1}\cdot e_i,\qquad e_i=1+\lambda(2w_i-1),
\]
with default \(\lambda=0.5\), and commits when \(E_t\ge 1/\alpha\). Under the null condition
\[
P(w_i=1\mid \mathcal{F}_{i-1})\le 1/2,
\]
the process is a nonnegative supermartingale, so
\[
P_{H_0}(\tau<\infty)\le \alpha.
\]
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 [2606.08106]. 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
\[
\delta_k=\frac{\delta_0}{Z\,k\log^2(k+1)},\qquad Z\approx 3.39,\qquad \sum_{k\ge 1}\delta_k=\delta_0.
\]
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 [2607.00871]. 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 \(H_0\), on filtration measurability, and on the stopping-time model under which optional-stopping validity is asserted [2603.05335]. Aggregation is delicate because convex mixtures preserve feasibility but can move a martingale frontier point into the interior of the supermartingale cone [2603.05335]. Time-sensitive testing leaves open robust criteria for composite alternatives and multi-threshold objectives [2605.06521]. Anchored e-watermarking depends on anchor calibration and null independence assumptions [2602.17608]. Cross-filtration lifting incurs an unavoidable logarithmic insurance cost [2402.09698].

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 [2603.05335]. 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.

Source: https://www.emergentmind.com/topics/anytime-valid-admissibility