Anytime-valid confidence sequence for a bounded mean

What it takes and gives

Runs with

Python 3.10 or later

Entry points

hoeffding_sequence, betting_sequence, pointwise_wilson_sequence, coverage_simulation

Known answers

6 cases its tests reproduce

Limits and scope

Limits

Means of bounded observations only; the bounds must be known in advance. The Hoeffding sequence is conservative; the betting sequence is tighter but costs grid x n operations and is meant for up to a few thousand observations. Grid reporting rounds intervals outward by at most one cell. Guarantees need a constant conditional mean; under drift the sequence covers no single target.

Guarantees

hoeffding: P(mu leaves the interval at some t) <= alpha, by Hoeffding's inequality at level alpha / (t (t + 1)) and a union bound. betting: the averaged capital (K+ + K-) / 2 at the true mean is a non-negative martingale, so P(it ever reaches 1 / alpha) <= alpha by Ville's inequality; the reported intervals contain every candidate mean the capital never rejects. Running intersections keep the simultaneous guarantee; empty_at records a time when the intersection would have become empty.

Assumes

Observations lie in the stated bounds and share one conditional mean given the past (for example independent and identically distributed). The level alpha is fixed before looking at the data.

Where it is shown

• Statistics for comparing resultsDeterministic skills

Related components

• Brier score decomposition and expected calibration error
• Split conformal prediction intervals
• Probabilistic and deflated Sharpe ratio
• Ledoit-Wolf covariance shrinkage
• Maximum mean discrepancy with an RBF kernel
• Weighted Vendi score