e-processes

Why This Matters

An e-value is a single nonnegative statistic with $\mathbb{E}_{H_0}[E] \leq 1$. An e-process is the sequential version: a collection $(E_t)_{t \geq 0}$ where each $E_t$ is an e-value with respect to the information available at time $t$, and the construction is consistent across times so that the value at any stopping time $\tau$ is still an e-value. The construction lets a tester accumulate evidence, peek as often as they like, stop whenever the evidence is convincing, and retain the same Type I error guarantee they would have had at a single fixed sample size.

The technical bridge is one of the cleanest results in modern statistics. If $E_t$ is a conditional e-value at time $t$ given the past, the running product $\prod_{s=1}^t E_s$ is a nonnegative supermartingale under the null. Ville's inequality bounds the maximum of a nonnegative supermartingale: $\Pr_{H_0}(\sup_t \prod E_t \geq 1/\alpha) \leq \alpha$. The supremum is over all times, including any stopping time the analyst might choose adaptively. Optional stopping, optional continuation, and data-dependent decision rules all preserve the bound.

This is the technical content of "anytime-valid inference." Every confidence sequence, every always-valid p-value, every safe sequential test in the modern literature is a special case of the e-process construction, dating back to Wald's 1945 SPRT (likelihood-ratio e-process between two simple hypotheses) and reaching forward to LLM evaluation pipelines and continuous-monitoring clinical trials.

Formal Setup

Definition

Filtration$({\mathcal{F}}_t)$

A nondecreasing family of $\sigma$-algebras $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \cdots$ on a probability space, with $\mathcal{F}_t$ representing the information available at time $t$. In practice, $\mathcal{F}_t = \sigma(X_1, \ldots, X_t)$ for an observed sequence.

Definition

Adapted process

A stochastic process $(Y_t)$ is adapted to $(\mathcal{F}_t)$ if $Y_t$ is $\mathcal{F}_t$-measurable for every $t$. Equivalently, $Y_t$ is a function of the observations through time $t$ only.

Definition

e-process$(E_t)$

A nonnegative process $(E_t)_{t \geq 0}$ adapted to the filtration $(\mathcal{F}_t)$, with $E_0 = 1$, such that for every stopping time $\tau$ (possibly infinite), $\sup_{P \in H_0} \mathbb{E}_P[E_\tau \cdot \mathbf{1}\{\tau < \infty\}] \leq 1.$ A test that rejects $H_0$ when $E_\tau \geq 1/\alpha$ has Type I error at most $\alpha$, by Markov's inequality applied to $E_\tau$.

Definition

Stopping time

A random variable $\tau : \Omega \to \{0, 1, \ldots\} \cup \{\infty\}$ such that the event $\{\tau \leq t\}$ is $\mathcal{F}_t$-measurable. Intuitively, the decision to stop at time $t$ depends only on the information available by time $t$.

Construction From Conditional e-values

The standard recipe builds an e-process from a sequence of conditional e-values. The constraint is that each new term has unit conditional expectation given the past.

Theorem

Running Product of Conditional e-values Is a Supermartingale

Statement

Let $(B_t)_{t \geq 1}$ be a sequence of nonnegative random variables adapted to $(\mathcal{F}_t)$ with $\mathbb{E}_{H_0}[B_t \mid \mathcal{F}_{t-1}] \leq 1$ for every $t$. Define $E_0 = 1$ and $E_t = E_{t-1} \cdot B_t$ for $t \geq 1$. Then $(E_t)$ is a nonnegative supermartingale under $H_0$ adapted to $(\mathcal{F}_t)$, and $\mathbb{E}_{H_0}[E_t] \leq 1$ for every $t$.

Consequently, $(E_t)$ is an e-process, and for every stopping time $\tau$, $\mathbb{E}_{H_0}[E_\tau \cdot \mathbf{1}\{\tau < \infty\}] \leq 1.$

Intuition

Each $B_t$ is a conditional bet: a payoff function with unit expected value given everything seen so far. Wealth multiplies across rounds. The conditional unit-expectation property is the martingale-like constraint that survives compounding. Equality at every step gives a martingale; an inequality gives a supermartingale, which is fine because we only need an upper bound on expected wealth.

Proof Sketch

Compute the conditional expectation of $E_t$ given $\mathcal{F}_{t-1}$: $\mathbb{E}_{H_0}[E_t \mid \mathcal{F}_{t-1}] = \mathbb{E}_{H_0}[E_{t-1} B_t \mid \mathcal{F}_{t-1}] = E_{t-1} \cdot \mathbb{E}_{H_0}[B_t \mid \mathcal{F}_{t-1}] \leq E_{t-1}.$ The factoring of $E_{t-1}$ out of the conditional expectation uses $\mathcal{F}_{t-1}$-measurability. The supermartingale property is the displayed inequality. The unconditional bound follows from the tower property: $\mathbb{E}_{H_0}[E_t] = \mathbb{E}_{H_0}[\mathbb{E}_{H_0}[E_t \mid \mathcal{F}_{t-1}]] \leq \mathbb{E}_{H_0}[E_{t-1}] \leq \cdots \leq 1$.

For the stopping-time statement, the optional stopping theorem (Doob) for nonnegative supermartingales gives $\mathbb{E}_{H_0}[E_\tau \mathbf{1}\{\tau < \infty\}] \leq \mathbb{E}_{H_0}[E_0] = 1$ for any stopping time $\tau$.

Why It Matters

This construction is the engine of every modern sequential test. Wald's SPRT is the special case where $B_t = p_1(X_t)/p_0(X_t)$ (the likelihood ratio of one observation), and the running product is the joint likelihood ratio. Confidence sequences are built from running products of inverted-bet e-processes. The Sequential Hypothesis Testing pipeline at large-scale A/B platforms (Optimizely, Statsig) reduces to this theorem applied to bounded-mean estimators.

Failure Mode

The bound requires the conditional unit-expectation property under the null. If the analyst chooses $B_t$ based on data not in $\mathcal{F}_{t-1}$ (e.g. by looking ahead, or by retrospectively reordering observations), the construction fails. The "adapted to $\mathcal{F}_{t-1}$" requirement is non-negotiable. Reusing the same observations across multiple $B_t$ definitions also breaks the conditional expectation, even though the resulting process is still adapted.

report a correction →

Optional ProofVille's inequality for nonnegative supermartingalesShowHide

The single-shot Markov inequality for an e-value generalizes to a maximal inequality for the entire e-process trajectory.

Theorem

Ville's Inequality

Statement

Let $(E_t)_{t \geq 0}$ be a nonnegative supermartingale under $P$ with $E_0 \leq 1$. Then for every $c > 0$, $P\!\left[\sup_{t \geq 0} E_t \geq c\right] \leq \frac{1}{c}.$ Setting $c = 1/\alpha$ gives the anytime-valid Type I error guarantee: under the null, the probability that the e-process ever exceeds $1/\alpha$ is at most $\alpha$.

Proof Sketch

Define $\tau = \inf\{t \geq 0 : E_t \geq c\}$, a stopping time. On the event $\{\sup_t E_t \geq c\}$, $\tau < \infty$ and $E_\tau \geq c$. By the optional stopping theorem, $\mathbb{E}[E_\tau \mathbf{1}\{\tau < \infty\}] \leq \mathbb{E}[E_0] \leq 1.$ On the complement, the bound is trivial. Apply Markov: $P(\tau < \infty) \cdot c \leq \mathbb{E}[E_\tau \mathbf{1}\{\tau < \infty\}] \leq 1,$ so $P(\sup_t E_t \geq c) = P(\tau < \infty) \leq 1/c$.

Why It Matters

The supremum is over all times. Ville's inequality says optional stopping at any time, including the time that maximizes the e-process value, cannot inflate Type I error beyond $\alpha$. This is the technical content of "anytime-valid."

Failure Mode

Ville's inequality requires the uniform supermartingale property: $\mathbb{E}[E_t \mid \mathcal{F}_{t-1}] \leq E_{t-1}$ for every $t$, not just $\mathbb{E}[E_t] \leq 1$ unconditionally. Mixing e-values from different filtrations or recomputing the conditional expectation only at certain times breaks the bound.

report a correction →

Canonical Example: Wald's SPRT

The sequential probability ratio test (Wald 1945) is the prototype e-process. Test $H_0: P_0$ against $H_1: P_1$, both simple, observe $X_1, X_2, \ldots$ iid. Define $E_t = \prod_{i=1}^t \frac{p_1(X_i)}{p_0(X_i)}.$ Each likelihood ratio is a conditional (in fact unconditional, by independence) e-value: $\mathbb{E}_{P_0}[p_1(X_i)/p_0(X_i)] = 1$. The running product is a nonnegative supermartingale under $H_0$ (actually a martingale, since the inequality is an equality).

Wald's original test stops the first time $E_t \geq A$ (reject $H_0$) or $E_t \leq B$ (accept $H_0$), with $A = 1/\alpha$ and $B = \beta$ for desired error rates. The Type I error guarantee follows from Ville: under $H_0$, $\Pr(\sup_t E_t \geq A) \leq 1/A = \alpha$. The expected sample size to reject under $H_1$ scales like $\log(1/\alpha)/\mathrm{KL}(P_1 \| P_0)$, the Kullback-Leibler rate.

The same construction works for composite nulls via universal inference (Wasserman-Ramdas-Balakrishnan 2020): plug in any density estimator for $P_1$ on a held-out half of the data, evaluate the likelihood ratio on the other half. The result is an e-process for any composite null.

Worked Exercise

ExerciseAdvanced

Problem

Let $X_1, X_2, \ldots$ be iid $\mathcal{N}(\theta, 1)$ and consider testing $H_0: \theta = 0$ against $H_1: \theta = \mu$ for fixed $\mu > 0$. Construct the likelihood-ratio e-process explicitly. Compute its value at $t = 100$ when $\bar X_{100} = 0.21$ and $\mu = 0.2$. At what time does the e-process exceed $1/\alpha = 20$ if the true mean is exactly $\mu = 0.2$ and $\bar X_t \approx 0.2$ for all $t$?

Hint

Reveal Solution

Practical Example: Hourly A/B Test Monitoring

A standard A/B test compares conversion rates of two website variants over a planned $N$ days of traffic. The product team wants to peek hourly and stop early if the result is convincing. With classical $z$-test $p$-values, peeking inflates Type I error. With an e-process:

1. Define $H_0: \text{conversion rate of variant A} \leq \text{conversion rate of variant B}$.
2. For each hour $t$, the streaming users contribute conditional bets $B_t$ based on the observed click-rate differential and the running variance estimate. Empirical-Bernstein constructions (Howard-Ramdas-McAuliffe-Sekhon 2021) give explicit formulas for $B_t$ with bounded outcomes.
3. The running product $E_t = \prod_{s \leq t} B_s$ is the e-process. Stop the test at the first $t$ where $E_t \geq 20$ (for $\alpha = 0.05$) or at the budget $T_{\max}$, whichever comes first.

Ville's inequality guarantees that under $H_0$, the probability of falsely rejecting at any time is at most $5\%$, regardless of how many hourly looks the analyst performed. Confidence sequences (the next topic) provide the matching interval estimates.

Connection to Classical Tests

Every classical sequential test maps onto an e-process construction:

• Wald's SPRT: likelihood-ratio e-process with two-sided thresholds.
• Group-sequential tests (Pocock, O'Brien-Fleming): e-process is implicit; the boundaries are tuned to the fixed schedule of looks. Adaptive peeking outside the schedule still breaks them; the e-process framing handles arbitrary stopping.
• Always-valid p-values: defined as $1/\sup_{s \leq t} E_s$, which inherits Ville's inequality.
• Bayes factors: under a prior, the Bayes factor is a likelihood ratio averaged over the prior, which is again an e-value. The product form across observations gives an e-process.

The modern view (Ramdas, Grünwald, Vovk, Shafer 2023) is that all sequential tests are e-processes under different choices of betting strategy.

Implementation Note

The standard implementation pattern is to maintain the log-e-process for numerical stability:

log_E = 0.0
for x_t in stream:
    log_B_t = compute_log_conditional_bet(x_t, history)
    log_E += log_B_t
    if log_E >= np.log(1 / alpha):
        return "reject H_0", t
    history.append(x_t)

The confseq Python package (Howard et al. 2021) provides ready-made empirical-Bernstein e-processes for bounded outcomes. For Gaussian observations with unknown variance, the Robbins (1970) mixture e-process is standard; for two-sample binomial tests, the betting construction from Waudby-Smith and Ramdas (2024) gives the tightest confidence sequences in the bounded-outcome regime.

Care must be taken with the filtration. Each $B_t$ must depend only on observations strictly before time $t$ (or, by convention, on $X_1, \ldots, X_{t-1}$ and any external randomness independent of $X_t$). Using the current observation $X_t$ inside $B_t$ violates adaptation and breaks the supermartingale property.

References

Canonical:

• Ramdas, A., Grünwald, P., Vovk, V., and Shafer, G. (2023). "Game-theoretic statistics and safe anytime-valid inference." Statistical Science 38(4), pp. 576-601. The modern survey, with Section 3 on e-processes and Section 4 on construction methods.
• Shafer, G. and Vovk, V. (2019). Game-Theoretic Foundations for Probability and Finance (Wiley). Chapters 3-7 develop the betting/martingale framework that e-processes formalize.
• Howard, S. R., Ramdas, A., McAuliffe, J., and Sekhon, J. (2021). "Time-uniform Chernoff bounds via nonnegative supermartingales." Probability Surveys 18, pp. 257-317. Quantitative time-uniform concentration inequalities derived from e-processes; the reference for empirical-Bernstein constructions.

Historical:

• Wald, A. (1945). "Sequential tests of statistical hypotheses." Annals of Mathematical Statistics 16(2), pp. 117-186. The SPRT.
• Wald, A. (1947). Sequential Analysis (Wiley). Book-length treatment of the SPRT and its operating characteristics.
• Ville, J. (1939). Étude critique de la notion de collectif (Gauthier-Villars). The original supermartingale maximal inequality, in a foundations-of-probability context.

Current:

• Waudby-Smith, I. and Ramdas, A. (2024). "Estimating means of bounded random variables by betting." Journal of the Royal Statistical Society, Series B 86(1), pp. 1-27. The state-of-the-art betting construction for confidence sequences.
• Grünwald, P., de Heide, R., and Koolen, W. (2024). "Safe testing." Journal of the Royal Statistical Society, Series B. e-process construction via reverse information projection for composite nulls.

Next Topics

• Anytime-valid inference: the broader framing of inference under continuous monitoring.
• Confidence sequences: time-uniform interval estimates derived from e-processes.
• Safe testing: the formal framework that builds tests directly on e-processes.
• E-values and anytime-valid inference: the umbrella reference with proofs and multiple-testing applications.
• Martingale theory: the mathematical foundation underneath the supermartingale construction.