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Betting confidence sequences for bounded random variables

Mathematical Methods AIMech:BettingConfidenceSequencesForBoundedMeans DRAFT

Aliases: Hedged Capital Process Confidence Sequence (Hedged-CS), Predictable Plug-in Empirical Bernstein Confidence Sequence (PrPl-EB), Approximate GRAPA Betting Martingale (aGRAPA), Hedged Grid Kelly Confidence Sequence (hgKelly), Betting Without Replacement Confidence Sequence (Hedged-WoR)

A nonparametric, nonasymptotic sequential and fixed-sample statistical inference framework (Waudby-Smith & Ramdas 2024) for estimating the common conditional mean mu in [0, 1] of a dependent sequence of [0, 1]-bounded random variables (X_t)_{t>=1} satisfying E_P[X_t | F_{t-1}] = mu by inverting composite family of test martingales (capital processes) K_t(m) = prod_{i=1}^t (1 + lambda_i(m)(X_i - m)) driven by F_{t-1}-predictable betting fractions lambda_t(m) in (-1/(1-m), 1/m) (Equation 14), or predictable plug-in Hoeffding and empirical Bernstein test supermartingales (Proposition 1, Theorem 2), so that by Ville's inequality the sublevel set C_t = {m in [0, 1] : K_t(m) < 1/alpha} forms a time-uniform (1 - alpha)-confidence sequence satisfying sup_{P in P^mu} P(exists t >= 1 : mu not in C_t) <= alpha and P(mu not in C_tau) <= alpha at any stopping time tau (Theorem 1).

Identity Notes & Concept Scope No normalized identity match for Betting confidence sequences for bounded random variables across all 442 pinned AIO records and 574 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only lines in Round 179 discovery artifacts where Waudby-Smith & Ramdas (2024, arXiv:2010.09686) was queued as a pending frontier lead. Distinct from AIMech:EDetectorSequentialChangeDetection (Shin, Ramdas, & Rinaldo 2024), which constructs cumulative-sum (e-SR) and running-maximum (e-CUSUM) e-detectors to detect an unknown changepoint with Average Run Length control E[N^*] >= 1/alpha, and from AIMech:UniversalInferenceSplitLikelihoodRatioTest (Wasserman, Ramdas, & Balakrishnan 2020) and AIMech:EValueCalibrationAndMerging (Vovk & Wang 2021), which do not develop predictable plug-in empirical Bernstein supermartingales, hedged capital processes K_t^pm(m), aGRAPA/LBOW Kelly approximations, or finite-population without-replacement betting confidence sequences.

Claims & Grounded Evidence

4 Claims

Mechanism #predictable-plugin-empirical-bernstein-supermartingale-and-asymptotic-variance-adaptivity

For any [0, 1]-valued sequence with E_P[X_t | F_{t-1}] = mu and any predictable sequence lambda_t in (0, 1), defining the predictable empirical mean mu_hat_t = (1/2 + sum_{i=1}^t X_i)/(t + 1) and variance sigma_hat_t^2 = (1/4 + sum_{i=1}^t (X_i - mu_hat_i)^2)/(t + 1) with psi_E(lambda) = (-log(1 - lambda) - lambda)/4 yields a non-negative test supermartingale prod_{i=1}^t exp(lambda_i(X_i - mu) - 4(X_i - mu_hat_{i-1})^2 psi_E(lambda_i)) via Fan et al.'s inequality exp(xi lambda - 4 xi^2 psi_E(lambda)) <= 1 + xi lambda for xi >= -1 (Proposition 2), generating the closed-form Predictable Plug-in Empirical Bernstein (PrPl-EB) confidence sequence C_t^PrPl-EB = (sum_{i=1}^t lambda_i X_i / sum_{i=1}^t lambda_i) +- (log(2/alpha) + 4 sum_{i=1}^t (X_i - mu_hat_{i-1})^2 psi_E(lambda_i)) / sum_{i=1}^t lambda_i (Theorem 2), whose fixed-n half-width W_n under i.i.d. variance sigma^2 scales asymptotically as sigma sqrt(2 log(2/alpha) / n) almost surely when lambda_i = sqrt(2 log(2/alpha) / (n sigma_hat_{i-1}^2)) wedge c (Proposition 9).

supports Source: Estimating means of bounded random variables by betting · Locator: Section 3.2, lines 183–249, Equations 10–13, Theorem 2; Section A.1, lines 1347–1384, Proposition 2 and Proof of Theorem 2; Section E.3, lines 1860–1901, Lemmas 6–8 and Proposition 9

Mechanism #hedged-capital-process-quasiconvexity-and-kelly-betting-hierarchies

For m-independent predictable pre-truncation bets lambda_dot_t^+, lambda_dot_t^- >= 0 truncated to lambda_t^+(m) = min(lambda_dot_t^+, c/m) and lambda_t^-(m) = min(lambda_dot_t^-, c/(1 - m)) with c in [0, 1) and allocation weight theta in [0, 1], the positive capital K_t^+(m) = prod_{i=1}^t (1 + lambda_i^+(m)(X_i - m)) is nonincreasing and log-convex on the active region, the negative capital K_t^-(m) = prod_{i=1}^t (1 - lambda_i^-(m)(X_i - m)) is nondecreasing, and both the hedged maximum K_t^pm(m) = max(theta K_t^+(m), (1 - theta) K_t^-(m)) and mixture martingale M_t^pm(m) = theta K_t^+(m) + (1 - theta) K_t^-(m) are quasiconvex in m so that their (1 - alpha)-sublevel sets B_t^pm = {m in [0, 1] : K_t^pm(m) < 1/alpha} and C_t^pm = cap_{i=1}^t B_i^pm are guaranteed connected intervals (Theorem 3, Lemma 2, and Section A.2); whereas the Hedged Grid Kelly process K_t^hgKelly(m) (Table 1, Proposition 5) is globally convex in m, unhedged m-dependent aGRAPA capital K_t^aGRAPA(m) can exhibit disconnected non-interval sublevel sets (Section E.4 counterexample).

supports Source: Estimating means of bounded random variables by betting · Locator: Section 4.1–4.3, lines 281–374, Equations 15–19, Theorem 3; Section B, lines 1425–1553, Table 1, Proposition 5 (Convexity of hgKelly); Section A.2, lines 1386–1416, Lemma 2 and Proof of Theorem 3; Section E.4, lines 1902–1908

Mechanism #universal-law-of-the-iterated-logarithm-rate-and-owens-empirical-likelihood-duality

Any Hoeffding factor exp(lambda(x - m) - lambda^2/8) with lambda in [-4/5, 4/5] is pointwise dominated on x, m in [0, 1] by a linear betting factor 1 + gamma^m(lambda)(x - m) with gamma^m(lambda) = exp(lambda(1 - m) - lambda^2/8) - exp(-lambda m - lambda^2/8) in [-1, 1] (Proposition 6), which implies via stitched geometric-grid mixtures that a discrete mixture of linear betting martingales achieves the universal law-of-the-iterated-logarithm shrinkage rate nu(C_t^L) = O(sqrt(log log t / t)) uniformly over all P in P^mu (Proposition 7) and O(1/sqrt(n)) fixed-sample width (Proposition 8); moreover, the optimal hindsight capital K_t^HS(m) = sup_{lambda in [-1/(1-m), 1/m]} prod_{i=1}^t (1 + lambda(x_i - m)) equals the exact reciprocal 1 / R(x_1^t; m) of Owen's (2001) nonparametric empirical likelihood ratio at lambda^EL = lambda^HS (Section E.6).

supports Source: Estimating means of bounded random variables by betting · Locator: Section E.1, lines 1732–1773, Proposition 6; Section E.2.1–E.2.2, lines 1774–1826, Propositions 7 and 8; Section E.6, lines 1914–1920 (Owen's empirical likelihood ratio duality)
supports Source: E-values: Calibration, combination, and applications · Locator: Section 2, lines 139–196 (betting capital interpretation of e-variables and test martingales)

Application #finite-population-sampling-without-replacement-hedged-betting-confidence-sequences

For sequential sampling without replacement (WoR) from any finite population {x_1, ..., x_N} subset [0, 1]^N with mean mu = (1/N) sum_{i=1}^N x_i, replacing the static null mean m at step t by the predictable remaining population mean m_t^WoR = (N m - sum_{i=1}^{t-1} X_i)/(N - t + 1) in the hedged capital factors K_t^+(m) = prod_{i=1}^t (1 + lambda_i^+(m)(X_i - m_i^WoR)) and K_t^-(m) = prod_{i=1}^t (1 - lambda_i^-(m)(X_i - m_i^WoR)) with truncation lambda_t^+(m) = min(lambda_dot_t^+, c / m_t^WoR) and lambda_t^-(m) = min(lambda_dot_t^-, c / (1 - m_t^WoR)) yields an exact test martingale under H_0^WoR : mu = m (Proposition 4, Section A.3) whose (1 - alpha)-level set C_t^WoR forms a valid finite-population confidence sequence that collapses to the exact singleton {mu} at t = N (Theorem 4).

supports Source: Estimating means of bounded random variables by betting · Locator: Section 5, lines 385–446, Equations 20–24, Theorem 4 (Hedged-WoR CSs); Section A.3, lines 1417–1424, Proposition 4 and Proof of Theorem 4

⚠️ Curated Limitations & Negative Results

Constructing valid betting martingales K_t(m) = prod_{i=1}^t (1 + lambda_i(m)(X_i - m)) requires known deterministic a priori support bounds [a, b] (rescaled to [0, 1]) so that predictable bets satisfy lambda_t(m) in (-1/(1-m), 1/m) and keep capital strictly non-negative; unbounded or heavy-tailed observations without known support bounds cannot use linear betting factors without breaking non-negativity (Section 1.2 and Section 4.1).
Unlike the closed-form Predictable Plug-in Hoeffding (PrPl-H) and Empirical Bernstein (PrPl-EB) intervals, the Hedged Capital confidence sequence C_t^pm has no closed-form algebraic endpoints and requires root-finding or grid evaluation over m in [0, 1] (with per-step cost O(G) for grid size G, though monotonicity of K_t^+(m) and K_t^-(m) enables binary search; Section 4.3 and Section B.4), while unhedged m-dependent aGRAPA bets can violate quasiconvexity and produce disconnected non-interval sublevel sets in pathological low-prior-variance regimes (Section E.4).
In finite-population sampling without replacement (Hedged-WoR), whenever the candidate remaining mean m_t^WoR = (N m - sum_{i=1}^{t-1} X_i)/(N - t + 1) falls outside [0, 1] at step t, the null value m is deterministically falsified (K_t^WoR(m) = infinity), and the method requires knowing the exact finite population size N in advance (Section 5.1).
All 1,971 lines of Waudby-Smith & Ramdas (2024, arXiv:2010.09686) were read. Constructed numerical checks in research/discovery/betting-confidence-sequences-bounded-means-boundaries.py verify Proposition 2 / Theorem 2 exact betting martingale vs. PrPl-EB supermartingale and Fan's inequality, Theorem 3 / Lemma 2 / Proposition 5 Hedged quasiconvexity and hgKelly convexity vs. Section E.4 unhedged aGRAPA non-convex sublevel sets, Theorem 4 / Proposition 4 Hedged-WoR exact martingale and width collapse, and Proposition 6 Hoeffding dominance plus Section E.6 Owen's empirical likelihood ratio duality.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Betting confidence sequences guarantee time-uniform coverage P(exists t >= 1 : mu not in C_t) <= alpha and valid coverage at arbitrary data-dependent stopping times tau via Ville's inequality on composite test martingales, preventing false-positive inflation from continuous sequential monitoring and optional stopping (AIO:DataDredgingBias).
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	Betting confidence sequences sequentially track and bound the common conditional mean E_P[X_t \| F_{t-1}] = mu of dependent bounded processes and finite-population sequential samples without i.i.d. assumptions, providing sequential uncertainty quantification for time-indexed streams (AIO:TimeSeriesAnalysis).

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`waudby-smith-ramdas2024-betting`	Estimating means of bounded random variables by betting Ian Waudby-Smith, Aaditya Ramdas	primary paper full_text	ID: https://doi.org/10.1093/jrsssb/qkad009 sha256:c41bcb5ff7e3…
`shin-ramdas-rinaldo2024-edetectors`	E-detectors: A nonparametric framework for sequential change detection Jaehyeok Shin, Aaditya Ramdas, Alessandro Rinaldo	primary paper full_text	ID: https://doi.org/10.51387/23-NEJSDS51 sha256:0c9c5ade08a8…
`vovk-wang2021-evalues`	E-values: Calibration, combination, and applications Vladimir Vovk, Ruodu Wang	primary paper full_text	ID: https://doi.org/10.1214/20-AOS2020 sha256:9ca237ebcb82…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T12:05:00Z

Updated: 2026-10-17T12:05:00Z

Source File: data/curated/BettingConfidenceSequencesForBoundedMeans.yaml · Schema: curated_concept.yaml

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