Universal inference split likelihood ratio test
Mathematical Methods AIMech:UniversalInferenceSplitLikelihoodRatioTest DRAFT
Aliases: Split Likelihood Ratio Test, Split LRT, Crossfit Likelihood Ratio Test, Universal Confidence Set via Split Likelihood Ratio Statistic, Running-MLE Sequential Likelihood Ratio Test
A regularity-free, finite-sample statistical hypothesis testing and confidence set construction framework (Wasserman, Ramdas, & Balakrishnan 2020) that partitions an observed sample into disjoint subsets D_0 and D_1, forms any out-of-sample alternative estimator or density hat_theta_1 on D_1 and the constrained null maximum likelihood estimator hat_theta_0 = argmax_{theta in Theta_0} L_0(theta) (or a proper upper-bounding relaxation F_0 >= L_0) on D_0, and rejects the composite null hypothesis H_0: theta^* in Theta_0 at level alpha in (0, 1) when the split likelihood ratio e-value U_n = L_0(hat_theta_1) / L_0(hat_theta_0) (or its two-fold crossfit average W_n = (U_n + U_n^swap)/2 or B-split subsampling average) exceeds 1/alpha, guaranteeing non-asymptotic Type I error at most alpha (Theorem 3) and (1 - alpha) coverage for the universal confidence set C_n = {theta in Theta : T_n(theta) <= 1/alpha} where T_n(theta) = L_0(hat_theta_1) / L_0(theta) (Theorem 1) via Markov's inequality without differentiability in quadratic mean or asymptotic chi-squared approximations.
Identity Notes & Concept Scope
No normalized identity match for Universal inference split likelihood ratio test across all 442 pinned AIO records and 572 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only lines in Round 177 discovery artifacts where Wasserman, Ramdas, & Balakrishnan (2020, arXiv:1912.11436) was queued as a pending frontier lead. Distinct from AIMech:EValueCalibrationAndMerging (Vovk & Wang 2021) and AIMech:EValueBenjaminiHochbergProcedure (Wang & Ramdas 2022), which combine or threshold pre-existing e-values across multiple hypotheses rather than constructing finite-sample composite-null e-values, confidence sets, and running-MLE confidence sequences for irregular statistical models via sample splitting, crossfitting, reversed information projections, and non-anticipating martingales.
Claims & Grounded Evidence
4 Claims
Mechanism
#finite-sample-validity-of-universal-confidence-sets-and-split-crossfit-lrt
Given an i.i.d. sample Y_1, ..., Y_{2n} ~ P_{theta^*} partitioned into disjoint subsets D_0 and D_1, let hat_theta_1 in Theta be any estimator computed solely from D_1 and L_0(theta) = prod_{i in D_0} p_theta(Y_i) be the out-of-sample likelihood on D_0; conditioning on D_1 fixes hat_theta_1 so that E_{theta^*}[T_n(theta^*) | D_1] = int_A prod_{i in D_0} p_{hat_theta_1}(y_i) d mu(y_i) <= 1 for T_n(theta) = L_0(hat_theta_1) / L_0(theta), which by Markov's inequality guarantees P_{theta^*}(theta^* in C_n) >= 1 - alpha for C_n = {theta in Theta : T_n(theta) <= 1/alpha} and its crossfit counterpart using S_n(theta) = (T_n(theta) + T_n^swap(theta))/2 (Theorem 1, Remark 2); for composite null testing H_0: theta^* in Theta_0, the null MLE hat_theta_0 = argmax_{theta in Theta_0} L_0(theta) satisfies L_0(hat_theta_0) >= L_0(theta^*), so U_n = L_0(hat_theta_1) / L_0(hat_theta_0) <= T_n(theta^*) has E_{theta^*}[U_n] <= 1 and E_{theta^*}[W_n] <= 1 for W_n = (U_n + U_n^swap)/2, controlling finite-sample Type I error at level alpha without regularity conditions (Theorem 3, Remark 4).
Mechanism
#dqm-parametric-and-hellinger-nonparametric-diameter-rates-and-gaussian-radius-scaling
In identifiable parametric models differentiable in quadratic mean (DQM) at theta^* with non-degenerate Fisher information I(theta^*), compact parameter space Theta subset R^d, and Lipschitz log-likelihood |log p_{theta_1}(x) - log p_{theta_2}(x)| <= ell(x) ||theta_1 - theta_2||, any sqrt(n)-consistent initial estimator hat_theta_1 yields a split-LRT confidence set C_n with diameter O_p(sqrt(log(1/alpha)/n)) (Theorem 5, Appendix A Lemmas 12–14); in nonparametric density classes P with Hellinger bracketing entropy integral bounded at rate epsilon_n (Eq. 11) and initial estimator hat_p_1 satisfying chi^2(p^*, hat_p_1) = O_p(eta_n^2) (Eq. 12), C_n has Hellinger diameter O_p(eta_n + epsilon_n + sqrt(log(1/alpha)/n)) (Theorem 6); and in the d-dimensional Gaussian model N_d(theta^*, I_d), C_n is a ball centered at Y_bar_0 with squared radius R_n^2 = ||Y_bar_0 - Y_bar_1||^2 + 2 log(1/alpha)/n ~ (2/n) chi^2_d + 2 log(1/alpha)/n (Eqs. 14–15), sharing the O_p(d/n) scaling of the classical LRT set A_n with asymptotic squared-radius ratio 4 as d -> infinity.
Mechanism
#support-mismatch-degeneracy-elimination-via-crossfitting-and-multi-split-e-averaging
For distributions with parameter-dependent support such as Uniform[0, theta] where hat_theta_1 = max_{i in D_1} Y_i and hat_theta_0 = max_{i in D_0} Y_i, the single-split likelihood L_0(hat_theta_1) = hat_theta_1^(-n) I(hat_theta_1 >= hat_theta_0) equals zero whenever hat_theta_1 < hat_theta_0 (which occurs with probability 1/2), causing C_n to degenerate to [0, infinity); in contrast, two-fold crossfitting S_n(theta) = (T_n(theta) + T_n^swap(theta))/2 ensures that exactly one of the two split statistics is nonzero for theta >= max(hat_theta_0, hat_theta_1), producing the rate-optimal bounded confidence interval [min(hat_theta_0, hat_theta_1), max(hat_theta_0, hat_theta_1)(2/alpha)^(1/n)] (or [max(hat_theta_0, hat_theta_1), max(hat_theta_0, hat_theta_1)(2/alpha)^(1/n)] after support intersection) of expected length O(theta log(1/alpha)/n) (Section 4), while averaging B split statistics T_bar_n = B^(-1) sum_{b=1}^B T_{n,b} across K-fold, repeated random subsampling, or all-splits partitions preserves E_{theta^*}[T_bar_n] <= 1 under arbitrary dependence between splits (Section 5).
Application
#profile-relaxed-powered-smoothed-ripr-extensions-and-running-mle-confidence-sequences
Universal inference extends to nuisance parameters via profile likelihoods L^dagger(psi) = sup_{theta in g^(-1)(psi)} L_0(theta), intractable nonconvex null MLEs via any proper upper bound F_0(theta) >= L_0(theta) on Theta_0 yielding T'_n = L_0(hat_theta_1)/F_0(hat_theta_0^F) <= T_n, robust inference via powered likelihoods L_0^eta (0 < eta < 1) via non-negativity of order-eta Renyi divergence, kernel-smoothed likelihoods via Jensen's inequality (Section 6), misspecified convex model classes via the reversed information projection (RIPR) p^* = argmin_{p in P_Theta} K(q, p) and the more powerful RIPR split LRT R_n = prod_{i in D_0} p_{hat_theta_1}(Y_i)/p_0^*(Y_i) >= U_n (Section 7, Propositions 7–8), nested sieve model selection without multiplicity correction (Section 4), and sequential anytime-valid p-values p_bar_t = min_{s <= t} 1/M_s and confidence sequences C_bar_t = cap_{s <= t} C_s via the running-MLE process M_t = prod_{i=1}^t p_{hat_theta_{1,i-1}}(Y_i) / prod_{i=1}^t p_{hat_theta_{0,t}}(Y_i) <= L_t bounded by the oracle nonnegative martingale L_t and Ville's inequality (Section 8, Theorems 9–11).
⚠️ Curated Limitations & Negative Results
- In regular parametric models where classical Wilks chi-squared asymptotics hold, universal inference is conservative due to sample splitting and Markov's inequality ('poor man's Chernoff bound' with exponent a = 1); in the d-dimensional Gaussian location model N_d(theta, I_d), the squared radius of the universal confidence ball C_n on 2n observations is approximately 4 times larger than that of the classical full-sample LRT confidence ball A_{2n} for large d (Section 2 and Section 3 Equations 13–15).
- Finite-sample validity of the split LRT (Theorem 3) and running-MLE sequential LRT (Theorem 9) requires that the denominator L_0(hat_theta_0) upper bound the true null likelihood L_0(theta^*); when the null likelihood is nonconvex (e.g., multi-component mixture models under H_0) and local optimizers such as EM fail to reach the global maximum, Type I error control is not guaranteed unless a certified convex or semidefinite upper bound F_0(theta) >= L_0(theta) is used (Section 4 and Section 6).
- Single-split universal inference suffers from algorithmic split randomness ('p-value lottery') and can degenerate to [0, infinity) with probability 1/2 when model densities have parameter-dependent support such as Uniform[0, theta], while the sequential running-MLE LRT requires recomputing hat_theta_{1,t-1} and hat_theta_{0,t} at each step t with O(t) denominator evaluation cost even in exponential families (Section 4, Section 5, and Section 8).
- All 1,098 lines of Wasserman, Ramdas, & Balakrishnan (2020, arXiv:1912.11436) were read. Constructed numerical checks in research/discovery/universal-inference-split-likelihood-ratio-test-boundaries.py verify Theorems 1 & 3 composite-null validity, Section 3 Gaussian squared-radius ratio 4.0, Section 5 Uniform[0, theta] crossfit interval vs. single-split degeneracy, Section 6 powered likelihood Renyi divergence identity, and Section 8 running-MLE martingale dominance M_t <= L_t.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Universal inference running-MLE sequential likelihood ratio tests, anytime-valid p-values (Theorem 10), and universal confidence sequences (Theorem 11) guarantee valid finite-sample Type I error and coverage under continuous peeking, optional stopping, optional continuation, and sequential sieve model selection without false-positive inflation from data dredging (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the selective-inference bias rather than the universal inference statistical methodology. |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
wasserman2020-universal-inference
|
Larry Wasserman, Aaditya Ramdas, Sivaraman Balakrishnan
|
primary paper
full_text
|
ID: https://doi.org/10.1073/pnas.1922664117
sha256:8788c234c31a…
|
vovk-wang2021-evalues
|
Vladimir Vovk, Ruodu Wang
|
primary paper
full_text
|
ID: https://doi.org/10.1214/20-AOS2020
sha256:9ca237ebcb82…
|
wang-ramdas2022-ebh
|
Ruodu Wang, Aaditya Ramdas
|
primary paper
full_text
|
ID: https://doi.org/10.1111/rssb.12489
sha256:26bd39717641…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T11:35:00Z
Updated: 2026-10-17T11:35:00Z
Source File:
data/curated/UniversalInferenceSplitLikelihoodRatioTest.yaml
· Schema:
curated_concept.yaml
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