TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Confidence sequences by test-supermartingale inversion

Statement

Assume AC and fix $α \in (0, 1)$ . Let $Θ$ be a parameter set, $(P_{θ})_{θ \in Θ}$ a family of laws on a common discrete filtered measurable space, and, for each $θ \in Θ$ , let $E^{θ} = (E_{t}^{θ})_{t \geq 0}$ be a nonnegative test supermartingale for the singleton null family ${P_{θ}}$ . Define $C_{t} (ω) = {θ \in Θ : E_{t}^{θ} (ω) < 1 / α} .$ Then $(C_{t})_{t \geq 0}$ is a $(1 - α)$ confidence sequence Confidence sequences.

No measurable structure on $Θ$ or joint measurability across $θ$ is needed for this pointwise membership and coverage statement.

Facts & Assumptions

Given: Assume AC, $α \in (0, 1)$ , the parameter family, and the candidate test-supermartingale $E^{θ}$ for each $θ$ . If $Θ$ is empty the conclusion is vacuous; otherwise fix an arbitrary $θ$ when proving its coverage bound.

[F1]

Nonnegative test supermartingales: for each fixed $θ$ , $E^{θ}$ is adapted and nonnegative under $P_{θ}$ .

[F2]

A test supermartingale gives anytime-valid p-values: transforming a test supermartingale by $p_{t} = 1$ when $E_{t} \leq 0$ and $p_{t} = \min (1, 1 / E_{t})$ otherwise gives an anytime-valid p-value process under its null law.

[F3]

Confidence sequences: each fixed-parameter membership event must be adapted and the true parameter must be covered simultaneously with probability at least $1 - α$ .

[A1]

The Axiom of Choice: AC is required by the conditional-expectation convention in the test-supermartingale definition and its validity theorem.

Proof

1.1F1F2A1given

Fix $θ \in Θ$ . Under $P_{θ}$ , define $p_{t}^{θ} = 1$ when $E_{t}^{θ} \leq 0$ and $p_{t}^{θ} = \min (1, 1 / E_{t}^{θ})$ otherwise. By adaptedness in F1 and the Borel transformation used in F2, each $p_{t}^{θ}$ is $F_{t}$ -measurable. Hence ${θ \in C_{t}} = {E_{t}^{θ} < 1 / α} = {p_{t}^{θ} > α}$ is $F_{t}$ -measurable.

2.1F2A1step 1.1

For this fixed $θ$ , F2 gives $P_{θ} (\exists t \geq 0 : p_{t}^{θ} \leq α) \leq α$ . The complement is exactly ${θ \in C_{t} for every t \geq 0}$ , whose probability is therefore at least $1 - α$ .

3.1F3step 1.1step 2.1∎

Steps 1.1 and 2.1 hold for every $θ$ . They give the membership measurability and simultaneous-coverage clauses in F3, so $(C_{t})$ is a confidence sequence.

Depends on

Used by

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Sources

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe and Jasjeet Sekhon, Time-Uniform, Nonparametric, Nonasymptotic Confidence Sequences, Lemma 3 and §6 (standard reference, not scraped)
Aaditya Ramdas, Peter Grunwald, Vladimir Vovk and Glenn Shafer, Game-Theoretic Statistics and Safe Anytime-Valid Inference, §2.8 (standard reference, not scraped)