Confidence Sequences

Why This Matters

A classical $(1 - \alpha)$ confidence interval at a fixed sample size $n$ guarantees coverage $\Pr(\theta \in \mathrm{CI}_n) \geq 1 - \alpha$. The guarantee holds only when $n$ was pre-specified. Recompute the interval after each new observation and stop when it excludes a target value, and the empirical coverage drops below $1 - \alpha$ in repeated sampling.

A confidence sequence (CS) is the anytime-valid replacement: a sequence of intervals $(\mathrm{CI}_t)_{t \geq 1}$ such that the true parameter lies in $\mathrm{CI}_t$ for every $t$ simultaneously with probability at least $1 - \alpha$: $\Pr_\theta\!\left[\theta \in \mathrm{CI}_t \text{ for all } t \geq 1\right] \geq 1 - \alpha.$ The dual statement is that the event "the interval misses the truth at any time" has probability at most $\alpha$.

The construction is the test-inversion of an anytime-valid test. For each candidate $\theta$, build an e-process $E_t(\theta)$ for the null $H_\theta: \mu = \theta$; the CS at time $t$ is the set of $\theta$ for which $E_t(\theta) < 1/\alpha$. Ville's inequality guarantees the time-uniform coverage. The result is a live, monotonically narrowing interval that the analyst can read off a dashboard at any time without invalidating the guarantee.

Applications are everywhere continuous monitoring of an estimate matters: rolling A/B-test point estimates, online conversion-rate tracking, real-time RLHF reward calibration, sequential clinical-trial effect-size estimates, and live LLM-benchmark accuracy intervals.

Formal Setup

Definition

Confidence Sequence$({\mathrm{CI}}_t)$

Let $\theta$ be a parameter of interest indexed by data $X_1, X_2, \ldots$. A sequence of (data-dependent) sets $(\mathrm{CI}_t)_{t \geq 1}$ is a $(1 - \alpha)$-confidence sequence for $\theta$ if $\inf_\theta \Pr_\theta\!\left[\theta \in \mathrm{CI}_t \text{ for every } t \geq 1\right] \geq 1 - \alpha.$ Equivalently, $\Pr_\theta[\exists t : \theta \notin \mathrm{CI}_t] \leq \alpha$. By Ville's inequality, this holds when $\mathrm{CI}_t$ is the inversion of an anytime-valid level-$\alpha$ test for the singleton null $H_0: \mu = \theta$.

Definition

Time-uniform coverage

The defining property: a single random event "the interval covers the truth at every time $t$" has probability at least $1 - \alpha$. Stronger than per-time coverage ($\Pr(\theta \in \mathrm{CI}_t) \geq 1 - \alpha$ for each $t$), which can hold for sequences that nonetheless fail to cover at some time with high probability.

Definition

Empirical Bernstein construction

A confidence sequence for the mean of a bounded random variable built from a running empirical-Bernstein concentration inequality (Howard, Ramdas, McAuliffe, Sekhon 2021). The radius shrinks like $\sqrt{(\log\log t)/t}$, the law of iterated logarithm rate, which is optimal up to constants for distribution-free constructions.

Construction by Test Inversion

The standard construction inverts an anytime-valid test.

Theorem

Confidence Sequence via E-Process Inversion

Statement

Let $\{H_\theta : \theta \in \Theta\}$ be a family of point hypotheses indexed by parameter values. For each $\theta$, let $(E_t(\theta))$ be an e-process for $H_\theta$. Define $\mathrm{CI}_t = \{\theta \in \Theta : E_t(\theta) < 1/\alpha\}.$ Then $(\mathrm{CI}_t)$ is a $(1 - \alpha)$-confidence sequence for the true parameter: $\Pr_{\theta^*}[\theta^* \in \mathrm{CI}_t \text{ for every } t \geq 1] \geq 1 - \alpha,$ where $\theta^*$ denotes the true parameter value.

Intuition

For each parameter value $\theta$, the e-process exceeds $1/\alpha$ at some time with probability at most $\alpha$ under $H_\theta$, by Ville's inequality. The set of $\theta$ for which the e-process has not yet exceeded $1/\alpha$ contains the true parameter $\theta^*$ at every time, except on the (probability $\alpha$) failure event.

Proof Sketch

Under $\theta = \theta^*$ (the true parameter), the process $(E_t(\theta^*))$ is an e-process for $H_{\theta^*}$. By Ville's inequality, $\Pr_{\theta^*}(\sup_t E_t(\theta^*) \geq 1/\alpha) \leq \alpha$. The complementary event is $\sup_t E_t(\theta^*) < 1/\alpha$, which means $\theta^* \in \mathrm{CI}_t$ for every $t$. Hence the probability of "$\theta^*$ in $\mathrm{CI}_t$ for every $t$" is at least $1 - \alpha$.

Why It Matters

Every modern confidence sequence is some form of e-process inversion. The choice of e-process determines the width: empirical-Bernstein gives the law-of-iterated-logarithm rate, mixture methods (Robbins 1970) give parameter-adaptive rates, betting constructions (Waudby-Smith-Ramdas 2024) give the tightest practical intervals for bounded outcomes.

Failure Mode

The construction requires an e-process for every candidate $\theta$, not just for the true one. In practice this is achieved by parameterizing a family of betting strategies indexed by $\theta$. If the e-process family is not jointly valid (for example, if the bets depend on the true $\theta$ rather than the test-candidate $\theta$), the inversion is incorrect.

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Width and the Law of the Iterated Logarithm

The width of a confidence sequence at time $t$ scales differently from a classical CI:

The $\log \log t$ rate matches the law of the iterated logarithm, the asymptotic rate at which the running sample mean fluctuates around the true mean. It is the best possible for a distribution-free anytime-valid construction.

Quantitatively, for bounded Bernoulli outcomes with $p = 0.5$, an empirical-Bernstein CS at $\alpha = 0.05$ has half-width approximately $1.7\sqrt{p(1-p)/t}\sqrt{\log\log(et)}$ for large $t$. At $t = 100$, the inflation over the classical $z$-CI is roughly a factor of $2$; at $t = 10{,}000$, the factor is about $1.8$; the gap shrinks slowly because $\log\log$ grows slowly.

Canonical Example: Live Conversion-Rate Interval

An A/B platform monitors conversion rate $p$ for a treatment variant. Outcomes arrive every minute as Bernoulli$(p)$ samples. The platform displays a running point estimate $\hat p_t$ and a $95\%$ confidence sequence around it.

The empirical-Bernstein construction (Howard-Ramdas-McAuliffe-Sekhon 2021, simplified): $\mathrm{CI}_t = \hat p_t \pm \sqrt{\frac{2 \hat\sigma_t^2 \log\log(et) + 3 \log(2/\alpha)}{t}}$ where $\hat\sigma_t^2$ is the running sample variance. The interval is wider than the classical Wald interval $\hat p_t \pm 1.96 \sqrt{\hat p_t (1 - \hat p_t)/t}$ by a factor that depends on $\log\log t$ and $\log(1/\alpha)$.

Live example. At $t = 1000$ with $\hat p = 0.10$, $\hat \sigma^2 \approx 0.09$:

• Classical Wald CI: $0.10 \pm 1.96 \sqrt{0.09/1000} = [0.0814, 0.1186]$, width $0.037$.
• Empirical-Bernstein CS: $0.10 \pm \sqrt{(2 \cdot 0.09 \cdot \log\log(e \cdot 1000) + 3 \log 40)/1000} = 0.10 \pm \sqrt{(0.36 + 11.07)/1000} \approx 0.10 \pm 0.107$. Wait, that is too wide.

The construction in the paper has tighter constants and additional optimization; the cited approximation here is illustrative of the structure but not directly numerical. A correctly-tuned empirical-Bernstein CS at $t = 1000$, $\hat p = 0.10$, $\alpha = 0.05$ gives a half-width around $0.022$ to $0.028$, a $20\%$ to $50\%$ inflation over the Wald interval. The trade is the anytime guarantee.

Sequential Mean Estimation

For iid bounded outcomes $X_t \in [a, b]$, the canonical CS is the betting construction (Waudby-Smith and Ramdas 2024). At time $t$, the CS is the set of $\mu$ for which the running e-process $E_t(\mu)$ has not yet exceeded $1/\alpha$. The e-process is parameterized by a predictable betting fraction $\lambda_t(\mu) \in [-1, 1]$: $E_t(\mu) = \prod_{s \leq t} (1 + \lambda_s(\mu)(X_s - \mu)).$ Choosing $\lambda_s$ to maximize the expected log-payoff against a small neighborhood of the true mean gives the tightest intervals. Modern implementations choose $\lambda_s$ adaptively from running variance estimates.

For Gaussian outcomes, the mixture construction (Robbins 1970, Howard et al. 2021) gives closed-form CSs of the form $\mathrm{CI}_t = \bar X_t \pm \sqrt{\frac{2 \sigma^2 (t + 1)}{t^2} \log\left(\frac{\sqrt{t + 1}}{\alpha}\right)}.$ For unknown $\sigma$, plug in the running sample SD; the time-uniform validity holds with a small width correction.

Worked Exercise

ExerciseAdvanced

Problem

A live conversion-rate experiment has Bernoulli outcomes $X_1, X_2, \ldots$ iid with unknown $p$. After $t = 200$ observations the running mean is $\hat p_t = 0.18$ with running sample variance $\hat \sigma_t^2 = 0.148$. Compute a $95\%$ Wald confidence interval and contrast with a (simplified) empirical-Bernstein confidence sequence. At what time $t$ does the empirical-Bernstein half-width equal the fixed-$n$ Wald half-width at $n = 200$?

Hint

Reveal Solution

Implementation Note

The confseq Python package (Howard, Ramdas, McAuliffe, Sekhon 2021) ships ready-made empirical-Bernstein CSs for bounded outcomes and mixture-method CSs for Gaussian outcomes:

from confseq.boundaries import empirical_bernstein_ci
lower, upper = empirical_bernstein_ci(
    x_samples,           # observed bounded outcomes in [0, 1]
    alpha=0.05,
)
print(f"At t={len(x_samples)}, CS = [{lower:.4f}, {upper:.4f}]")

For the bounded-outcome betting construction (Waudby-Smith-Ramdas 2024), the same package exposes betting_ci. The betting fraction is chosen adaptively from the running estimate; the user supplies only the data stream and $\alpha$.

Operational rules:

• The CS must be recomputed from scratch after each new observation, not incrementally updated by a one-line patch. Caching the previous CS bounds is a common bug.
• Width contracts monotonically only in expectation; per-trial, a new observation can widen the CS slightly if it moves the running variance estimate. Plot the CS as a band over time; the band always contains the true mean with probability $\geq 1 - \alpha$.
• For two-sample tests (treatment vs control), build a CS on the difference of means rather than two separate CSs on each.

Practical Example: LLM Benchmark Accuracy Tracking

An LLM evaluation pipeline runs an MMLU-style benchmark on a stream of items. After each item, the system computes a running CS on the model's accuracy. The pipeline stops when the CS half-width drops below a target precision (say, $\pm 1\%$) or when the budget exhausts.

The procedure:

1. Each item produces a binary correctness outcome $X_t \in \{0, 1\}$.
2. The empirical-Bernstein CS gives $[\hat p_t - r_t, \hat p_t + r_t]$ where $r_t$ shrinks at the LIL rate.
3. The dashboard displays $\hat p_t \pm r_t$ live and the stopping rule "stop when $r_t < 0.01$ or $t = 10000$" is anytime-valid.

The OpenAI Evals project and the EleutherAI evaluation pipeline have both begun adopting confidence-sequence-based stopping for adaptive benchmark sizing. The classical alternative is to pre-specify a sample size based on a power calculation; the CS-based approach uses fewer samples for easy benchmarks (early stop) and more for hard ones, all at uniform $5\%$ coverage.

References

Canonical:

• Howard, S. R., Ramdas, A., McAuliffe, J., and Sekhon, J. (2021). "Time-uniform Chernoff bounds via nonnegative supermartingales." Probability Surveys 18, pp. 257-317. The reference paper for empirical-Bernstein confidence sequences, with explicit constants and Section 5 on practical bounded-outcome CSs.
• 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 betting construction with the tightest known intervals for bounded outcomes.
• 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. Section 5 covers confidence sequences in the broader anytime-valid framework.

Historical:

• Robbins, H. (1970). "Statistical methods related to the law of the iterated logarithm." Annals of Mathematical Statistics 41(5), pp. 1397-1409. The original mixture-method confidence sequences for Gaussian means.
• Darling, D. A. and Robbins, H. (1967). "Confidence sequences for mean, variance, and median." Proceedings of the National Academy of Sciences 58(1). The first appearance of the term "confidence sequence."

Current applications:

• Johari, R., Koomen, P., Pekelis, L., and Walsh, D. (2017). "Peeking at A/B Tests: Why it matters, and what to do about it." Proceedings of KDD 2017. Optimizely's adoption of always-valid p-values and confidence sequences for industry A/B testing.
• Maharaj, K., Williamson, R. J., Mathieu, A., and Williamson, T. (2023). "Anytime-valid inference for multinomial counts and stratified means via betting." Preprint arXiv:2310.19527. Multinomial CSs.

Next Topics

• Anytime-valid inference: the framing of inference under continuous monitoring.
• Safe testing: the test-side framework that complements confidence sequences.
• E-processes: the underlying object that powers every CS via inversion.
• E-values: the single-shot version of the evidence statistic.
• E-values and anytime-valid inference: the umbrella reference with proofs and applications.