A confidence sequence can be thought of as a sequence of sequentially valid confidence intervals). Formally, $(C_{t})_{t≥1}$ is a $(1−α)$-CS for a parameter $θ$ if

$P(∃t≥1:θ∈/C_{t})≤α.$That is, a CS is a time-uniform confidence interval. It avoids the known problem with confidence intervals that, if they are computed at different times, they risk contradicting themselves (eg an interval computed at time $t_{1}$ might not overlap with one at time $t_{2}$. )

Like confidence intervals, confidence sequences have frequentist guarantees (frequentist statistics), not Bayesian ones. Whether there exist anytime-valid credible intervals (the Bayesian notion of confidence interval) is an interesting question.

Equivalent definitions are:

- $P(θ_{∗}∈C_{τ})≥1−α$ for any stopping time $τ$
- $P(θ_{∗}∈C_{T})≥1−α$ for all random times $T$. A random time could be a function of a stopping time, for instance (eg $τ/2$).

These are not obviously equivalent definitions. They have to be proved. For instance, the same properties do not hold with expected values. See lemmas 2 and 3 here and the anytime-valid notes.

Confidence sequences are common tools in safe, anytime-valid inference (SAVI).

Many confidence sequences are construct from applying Ville’s inequality to supermartingales or test-martingales]. It is known how to construct non-trivial confidence sequences for a variety of problems. A few include:

- bounded, non-parametric mean estimation in $R$ (estimating means by betting) and in $R_{d}$ (CSs via universal gambling strategies), and in Banach spaces (via empirical Bernstein bounds).
- Mean estimation under light-tailed sub-psi conditions: See Howard et al. 2021.
- Mean estimation for heavy-tailed scalar observations (see discussion in Catoni-Giulini M-estimator) with possibly infinite variance.
- Estimating Gaussian means with unknown variance
- Light and heavy-tailed (finite variance) mean estimation in $R_{d}$, using the variational approach to concentration.
- confidence sequences for quantiles

There is also an extension of confidence sequences to the asymptotic regime: asymptotic confidence sequences.

# Refs

- Time-uniform, nonparametric, nonasymptotic confidence sequences by Howard et al.