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Sub-psi uniform boundary confidence sequences

Mathematical Methods AIMech:SubPsiUniformBoundaryConfidenceSequences DRAFT

Aliases: Time-Uniform Nonparametric Nonasymptotic Sub-Psi Confidence Sequences, Polynomial Stitched Sub-Gamma Uniform Boundary (S_alpha), Conjugate Mixture Sub-Psi Uniform Boundaries (Normal, Beta-Binomial, Gamma-Exponential, Gamma-Poisson), Discrete Mixture Sub-Psi Uniform Boundary (DM_alpha), Self-Normalized Empirical Bernstein Sub-Exponential Confidence Sequence

A general nonparametric, nonasymptotic sequential estimation and boundary-crossing framework (Howard, Ramdas, McAuliffe, & Sekhon 2021) for any pair of adapted processes (S_t, V_t)_{t>=0} satisfying the sub-psi condition exp(lambda S_t - psi(lambda) V_t) <= L_t(lambda) for a non-negative supermartingale L_t(lambda) with E[L_0(lambda)] <= l_0 and a CGF-like function psi : [0, lambda_max) -> R_>=0 (canonical sub-Bernoulli psi_B, sub-Gaussian psi_N, sub-Poisson psi_P, sub-exponential psi_E, and sub-gamma psi_G; Definitions 1–3, Proposition 1), constructing curved time-uniform boundaries u(V_t) via geometric epoch stitching (Theorem 1), analytical conjugate mixtures (Propositions 2–10), geometric-grid discrete mixtures (Theorem 2), and inverted stitching (Theorem 3) such that P(exists t >= 1 : S_t >= u(V_t)) <= alpha and the induced confidence sequence (C_t)_{t>=1} satisfies P(forall t >= 1 : theta_t in C_t) >= 1 - alpha equivalently at all arbitrary stopping times tau (Lemma 3).

Identity Notes & Concept Scope No normalized identity match for Sub-psi uniform boundary confidence sequences across all 442 pinned AIO records and 575 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only lines in Round 180 discovery artifacts where Howard, Ramdas, McAuliffe, & Sekhon (2021, arXiv:1810.08240) was queued as a pending frontier lead. Distinct from AIMech:BettingConfidenceSequencesForBoundedMeans (Waudby-Smith & Ramdas 2024), which constructs multiplicative linear betting capital processes K_t(m) = prod_{i=1}^t (1 + lambda_i(m)(X_i - m)) and predictable plug-in intervals for [0, 1]-bounded observations rather than sub-psi exponential supermartingale uniform boundaries u(V_t) over intrinsic variance time V_t across the five canonical sub-psi families (sub-Bernoulli, sub-Gaussian, sub-Poisson, sub-exponential, sub-gamma), polynomial epoch stitching S_alpha(v), conjugate beta-binomial/gamma-exponential/gamma-Poisson mixtures, geometric discrete mixtures DM_alpha(v), inverted stitching, and matrix finite-LIL eigenvalue bounds.

Claims & Grounded Evidence

4 Claims

Mechanism #sub-psi-cumulant-hierarchy-and-universal-sub-gamma-reduction

The five canonical CGF-like functions psi_{B,g,h}(lambda) = (1/(gh)) log((g e^{h lambda} + h e^{-g lambda})/(g+h)), psi_N(lambda) = lambda^2/2, psi_{P,c}(lambda) = c^{-2}(e^{c lambda} - c lambda - 1), psi_{E,c}(lambda) = c^{-2}(-log(1 - c lambda) - c lambda), and psi_{G,c}(lambda) = lambda^2 / (2(1 - c lambda)) satisfy the strict pointwise ordering psi_{B,1,1}(lambda) < psi_N(lambda) < psi_{P,1}(lambda) < psi_{E,1}(lambda) < psi_{G,1}(lambda) on lambda in (0, 1) together with the universal sub-Poisson-to-sub-gamma bound psi_{P,c}(lambda) <= psi_{G,c/3}(lambda) for all lambda in [0, 3/c) (Proposition 1), so every sub-psi process is sub-gamma (and sub-exponential) with scale parameter c' and any sub-gamma or sub-exponential uniform boundary applies universally across all five families with optimal variance-rescaling constants a given by Proposition 11 and Table 1.

supports Source: Time-uniform, nonparametric, nonasymptotic confidence sequences · Locator: Section 2, lines 135–218, Definitions 1–2, Equations 2–6, Proposition 1, Figure 2; Appendix B, lines 1336–1342, Proposition 11 and Table 1
supports Source: E-detectors: A nonparametric framework for sequential change detection · Locator: Section 5.1, lines 560–615, Table 2 (Canonical sub-psi CGF-like pair hierarchy)

Mechanism #polynomial-epoch-stitching-and-inverted-stitching-uniform-boundaries

Partitioning intrinsic variance time into geometric epochs [m eta^k, m eta^{k+1}) for eta > 1, m > 0 and intersecting tangent linear boundaries g_k(v) = r_k / lambda_k + (psi_{G,c}(lambda_k)/lambda_k) v tuned via the exact inverse psi_{G,c}^{-1}(u) = 2u / (c u + sqrt(2u + c^2 u^2)) at lambda_k = psi_{G,c}^{-1}(r_k / (m eta^{k+1/2})) achieves exact normalized endpoint symmetry g_k(m eta^k)/eta^{k/2} = g_k(m eta^{k+1})/eta^{(k+1)/2} and yields the closed-form sub-gamma stitched boundary S_alpha(v) = sqrt(k_1^2 v l(v) + c^2 k_2^2 l(v)^2) + c k_2 l(v) with k_1 = (eta^{1/4} + eta^{-1/4})/sqrt(2), k_2 = (sqrt(eta) + 1)/2, and l(v) = s log log(eta v / m) + log(zeta(s)/(alpha log^s eta)), growing at the iterated-logarithm rate O(sqrt(v log log v)) (Theorem 1, Appendix A.1 Equations 29–37); conversely, Theorem 3's inverted stitching bounds the sub-Gaussian upcrossing probability of any nonnegative strictly concave curve g(v) over [1, v_max] by sum_{k=0}^{lceil log_eta v_max rceil - 1} exp(-2 a_k b_k) using secant intercepts a_k and slopes b_k.

supports Source: Time-uniform, nonparametric, nonasymptotic confidence sequences · Locator: Section 3.1, lines 221–248, Equations 7–11, Theorem 1; Section 3.4, lines 278–286, Equations 18–20, Theorem 3; Appendix A.1, lines 1096–1129, Equations 29–37; Appendix A.7, lines 1298–1313, Lemma 8 and Proof of Theorem 3

Mechanism #conjugate-and-discrete-mixture-boundaries-with-lambert-w-optimal-tuning

Integrating exp(lambda S_t - psi(lambda) V_t) against a mixing distribution F produces a supermartingale M_t whose inverted boundary M_alpha(v) = sup{s : m(s, v) < l_0/alpha} is continuous, increasing, and concave (Lemma 9), achieves asymptotic rate sqrt(2 c v log(v)/2) when psi''(0+) = c > 0 (Proposition 2), and is unimprovable over 1-sub-Gaussian processes (Proposition 4), yielding closed-form or special-function conjugate boundaries for two-sided and one-sided normal mixtures (Propositions 5–6), beta-binomial mixtures (Propositions 7–8), gamma-exponential mixtures (Proposition 9), and gamma-Poisson mixtures (Proposition 10), where the normalized boundary u(v)/sqrt(v) of the two-sided normal mixture is uniquely minimized at intrinsic time m when the precision parameter satisfies m / rho = -W_{-1}(-alpha^2 / (e l_0^2)) - 1 via the lower Lambert W_{-1} branch (Proposition 3, Section C.1), and Theorem 2's geometric-grid discrete mixture DM_alpha(v) with lambda_k = lambda_bar / eta^{k+1/2} and w_k = lambda_bar (eta - 1) f(lambda_bar / eta^k) / eta^{k+1} conservatively upper-bounds exact continuous mixtures for arbitrary nonincreasing densities f.

supports Source: Time-uniform, nonparametric, nonasymptotic confidence sequences · Locator: Section 3.2–3.6, lines 249–328, Equations 12–17 and 21, Propositions 2–4, Theorem 2; Appendix A.2–A.5, lines 1130–1291, Lemma 2, Propositions 5–10, Lemmas 4–7; Appendix C.1–C.2, lines 1344–1375, Lemma 9

Application #self-normalized-empirical-bernstein-and-adaptive-ipw-ate-confidence-sequences

For any sequence X_t in [a, b] a.s. with average conditional expectation mu_t = t^{-1} sum_{i=1}^t E_{i-1}[X_i] and any [a, b]-valued F_{t-1}-predictable sequence X_hat_t, the centered sum S_t = sum_{i=1}^t (X_i - E_{i-1}[X_i]) is sub-exponential with scale c = b - a and empirical variance process V_t = sum_{i=1}^t (X_i - X_hat_i)^2 via Fan et al.'s inequality exp(lambda xi - psi_{E,1}(lambda) xi^2) <= 1 + lambda xi without taking a union bound over separate mean and variance concentration events, yielding the time-uniform empirical Bernstein confidence sequence P(forall t >= 1 : |X_bar_t - mu_t| < u(V_t)/t) >= 1 - 2 alpha (Theorem 4, Appendix A.8) that avoids the severe false-positive inflation of the naive self-normalized sub-Gaussian mixture (Equation 24) on skewed bounded distributions (Section 5, Figure 4a), and extends directly to sequential Neyman-Rubin average treatment effect estimation ATE_t under adaptive treatment assignment P_t in [p_min, 1 - p_min] with scale c = 2 / p_min (Corollary 2) as well as matrix martingale finite-LIL eigenvalue bounds (Corollaries 4–5).

supports Source: Time-uniform, nonparametric, nonasymptotic confidence sequences · Locator: Section 4.1–4.3, lines 329–395, Equations 22–23, Theorem 4, Corollaries 2–5; Section 5, lines 440–515, Equation 24, Figure 4a; Appendix A.8–A.10, lines 1315–1335

⚠️ Curated Limitations & Negative Results

No uniform boundary can be simultaneously tight across all intrinsic variance times V_t: conjugate mixture boundaries with positive density at the origin grow asymptotically at rate O(sqrt(v log v)) (Proposition 2) rather than the O(sqrt(v log log v)) iterated-logarithm rate of polynomial stitched (Theorem 1) and LIL discrete mixture boundaries (Theorem 2), trading asymptotic looseness as v -> infinity for tighter intervals over approximately three orders of magnitude around the Lambert-W-tuned target variance time m (Section 3.5, Proposition 3).
Except for the two-sided normal mixture (Proposition 5), the closed-form upper bound for the one-sided normal mixture (Proposition 6), and polynomial stitched sub-gamma boundaries (Theorem 1), evaluating conjugate mixture boundaries (beta-binomial Propositions 7–8, gamma-exponential Proposition 9, gamma-Poisson Proposition 10) and discrete mixture boundaries (Theorem 2) requires numerical root-finding over s at each step t together with incomplete Beta or incomplete Gamma special functions (Appendix D).
Theorem 4's self-normalized empirical Bernstein confidence sequence and Corollary 2's adaptive IPW ATE confidence sequence require known a priori support bounds X_t in [a, b] a.s. (and propensity scores P_t bounded away from 0 and 1 in [p_min, 1 - p_min]) to fix the sub-exponential scale parameter c = b - a (or c = 2 / p_min); naively substituting empirical variance V_t into a scale-free sub-Gaussian normal mixture (Equation 24) fails violently on heavily skewed bounded distributions (Section 5, Figure 4a).
All 1,696 lines of Howard, Ramdas, McAuliffe, & Sekhon (2021, arXiv:1810.08240) were read. Constructed numerical checks in research/discovery/sub-psi-uniform-boundary-confidence-sequences-boundaries.py verify the strict five-family sub-psi hierarchy and Proposition 1 sub-gamma reduction, Theorem 1 epoch stitching endpoint symmetry and dominance plus Theorem 3 inverted stitching, Propositions 3/5/6/9 Lambert W_{-1} tuning and gamma-exponential-to-normal convergence plus Theorem 2 discrete mixture accuracy, and Theorem 4 / Corollary 2 empirical Bernstein and IPW ATE coverage vs. Section 5's skewed-distribution naive self-normalized counterexample.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Sub-psi uniform boundary confidence sequences maintain nonasymptotic coverage P(forall t >= 1 : theta_t in C_t) >= 1 - alpha simultaneously across all times t >= 1 and arbitrary data-dependent stopping times tau (Lemma 3), preventing false-positive inflation from optional stopping and continuous peeking (AIO:DataDredgingBias).
`AIO:TimeSeriesAnalysis` Time Series Analysis	related_match	Sub-psi uniform boundaries track time-varying average conditional expectations mu_t = t^{-1} sum_{i=1}^t E_{i-1}[X_i], sequential causal treatment effects ATE_t, and martingale state-space/matrix processes over intrinsic variance time V_t without i.i.d. assumptions (AIO:TimeSeriesAnalysis).

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`howard-ramdas-mcauliffe-sekhon2021-subpsi`	Time-uniform, nonparametric, nonasymptotic confidence sequences Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, Jasjeet Sekhon	primary paper full_text	ID: https://doi.org/10.1214/20-AOS1991 sha256:aef352a6e271…
`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…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T12:20:00Z

Updated: 2026-10-17T12:20:00Z

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

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