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A Gaussian mixture confidence sequence for conditional sub-Gaussian means

Statement

Assume AC. Let $Θ \subseteq R$ , let $(Ω, F, (F_{t})_{t \in N_{0}})$ be a common discrete filtered measurable space, and let $(P_{θ})_{θ \in Θ}$ be a family of probability laws on it. Let $(X_{t})_{t \geq 1}$ be a common real-valued process that is adapted under each $P_{θ}$ , and suppose that for every $θ \in Θ$ its increments are conditionally sub-Gaussian around $θ$ with the same known proxy $σ^{2} > 0$ under $P_{θ}$ in the sense of Conditionally sub-Gaussian increments. Fix $ρ > 0$ and $α \in (0, 1)$ . For $t \geq 1$ , set ${\overset{ˉ}{X}}_{t} : = \frac{1}{t} \sum_{i = 1}^{t} X_{i}, C_{t} : = Θ \cap [{\overset{ˉ}{X}}_{t} - \frac{u_{α} (t σ^{2})}{t}, {\overset{ˉ}{X}}_{t} + \frac{u_{α} (t σ^{2})}{t}],$ where $u_{α}$ is from Gaussian-mixture confidence-sequence boundary, and set $C_{0} : = Θ$ . Then $(C_{t})_{t \geq 0}$ is a $(1 - α)$ confidence sequence for $θ$ : $P_{θ} (θ \in C_{t} for every t \geq 0) \geq 1 - α (θ \in Θ) .$ The claim is pointwise in $θ$ and requires no measurable structure on $Θ$ , independence, or identical distribution of the observations.

Facts & Assumptions

Given: Assume AC and all data, law, filtration, proxy, and tuning hypotheses in the Statement.

[F1]

Conditionally sub-Gaussian increments: for each fixed $θ$ , time $t \geq 1$ , and $λ \in R$ , the conditional mgf of $X_{t} - θ$ given $F_{t - 1}$ is at most $e^{λ^{2} σ^{2} / 2}$ almost surely.

[F2]

The Gaussian mixture is a sub-Gaussian test supermartingale: under each $P_{θ}$ , $M_{t} (θ) = \sqrt{ρ / (t σ^{2} + ρ)} \exp (S_{t} (θ)^{2} / [2 (t σ^{2} + ρ)])$ is a nonnegative test supermartingale with $M_{0} (θ) = 1$ .

[F3]

The Gaussian-mixture threshold is the stated square-root boundary: $M_{t} (θ) \geq 1 / α$ exactly when $∣ S_{t} (θ) ∣ \geq u_{α} (t σ^{2})$ .

[F4]

Discrete Ville inequality and anytime threshold validity: under AC, for every null law and each $α \in (0, 1)$ , a nonnegative test supermartingale has $P (\sup_{t \geq 0} M_{t} \geq 1 / α) \leq α$ .

[F5]

Confidence sequences: a $(1 - α)$ confidence sequence has $F_{t}$ -measurable membership events and covers each true parameter at all integer times with probability at least $1 - α$ .

[A1]

The Axiom of Choice: AC supplies the conditional-expectation and test-supermartingale conventions used in [F2] and [F4].

Proof

1.1givenF1F5

Fix $θ \in Θ$ . The process is adapted under $P_{θ}$ , so each $X_{i}$ for $i \leq t$ and hence ${\overset{ˉ}{X}}_{t}$ is $F_{t}$ -measurable. Therefore ${θ \in C_{t}}$ is $F_{t}$ -measurable for every $t \geq 1$ , and it is the whole space at $t = 0$ .

2.1givenF3step 1.1

For $t \geq 1$ , if $θ \notin C_{t}$ , the interval in the Statement has closed endpoints, so $∣ S_{t} (θ) ∣ > u_{α} (t σ^{2})$ . By [F3], this implies $M_{t} (θ) \geq 1 / α$ . Consequently the event that $θ$ is ever excluded from $C_{t}$ is contained in ${\sup_{t \geq 0} M_{t} (θ) \geq 1 / α}$ .

3.1F2F4F5A1step 2.1∎

By [F2] and Ville's inequality [F4], the latter event has $P_{θ}$ -probability at most $α$ . Its complement therefore has probability at least $1 - α$ and is contained in ${θ \in C_{t} for all t \geq 0}$ . Since [F5] requires exactly these membership and coverage clauses, $(C_{t})$ is a confidence sequence.

Depends on

Used by

Independent Gaussian increments satisfy the conditional proxy Example

Dependency tree · two levels

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Sources

Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform, nonparametric, nonasymptotic confidence sequences, Lemma 2, Proposition 5, and Appendix A.3 (standard reference, not scraped)
Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon, Time-uniform Chernoff bounds via nonnegative supermartingales, Theorem 1 and §2.3 (standard reference, not scraped)
Aaditya Ramdas, Martingales 1: Concentration Inequalities — Conjugate Mixtures, 36-771 lecture notes, Carnegie Mellon University, Fall 2018 (standard reference, not scraped)