PaperlayerView exact revision on arXiv

Source-linked AI summary

SLICE: Sequential Likelihood-Free Inference via Conformal E-Values

Stefano Cortinovis, Laura Battaglia, François Caron

arXiv:2610.13855v1stat.MEstat.ML

TL;DR

Likelihood-free models can be simulated even when their likelihoods are intractable, leaving frequentist uncertainty quantification often approximate or fixed-sample. SLICE normalizes predictable scores with fresh null simulations and inverts test martingales to form anytime-valid confidence sequences. Neural scores perform close to likelihood-ratio and application-specific scores in the reported benchmarks and applications.

  • Problem

    Likelihood-free inference often relies on uncertainty methods that are approximate or designed for a prespecified sample size, rather than valid under continuous monitoring.

    Source: §1 · p. 1

  • Method

    SLICE normalizes a positive predictable score using fresh simulations from each candidate null, multiplies the resulting conditional e-values into test martingales, and inverts them for confidence sequences.

    Source: §1 · p. 1, §4.1 · pp. 3–4

  • Results

    Neural scores yield confidence sequences close to conformalized likelihood ratios on synthetic benchmarks and perform comparably to application-specific scores in an LLM routing audit and microlensing inference.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Validity holds for any positive predictable score, while score choice determines efficiency; neural ratio scores are a practical approximation to the oracle likelihood-ratio score.

    Source: §1 · p. 1

    The stated coverage applies to the evaluation grid; coverage beyond it requires additional control between grid points.

    Source: §4.4 · p. 5

Abstract

from arXiv · show

Many scientific models can be simulated but have intractable likelihoods, making frequentist uncertainty quantification difficult. We construct exact anytime-valid confidence sequences for the parameters of such models. At each observation, fresh simulations from each candidate null conformalize a positive predictable score into a conditional e-value; their product is a test martingale, which we invert over the parameter. Validity holds for any such score, whatever its accuracy. We show that the oracle likelihood-ratio score maximizes expected log growth, which motivates approximating it with an amortized neural ratio estimator. On synthetic benchmarks, an audit of model routing in an LLM API, and microlensing data, the resulting confidence sequences stay valid under continuous monitoring, where fixed-sample methods do not, and neural scores remain competitive with exact likelihood ratios and application-specific scores.

1 INTRODUCTION

Likelihood-free models enable simulation when likelihoods are unavailable, but existing uncertainty methods are often approximate or fixed-sample. SLICE uses normalized scores and sequential test martingales to construct anytime-valid confidence sequences, with neural ratio scores motivated by an oracle efficiency result.

1 INTRODUCTION

  • SLICE targets anytime-valid frequentist uncertainty quantification when likelihoods are unavailable, unlike methods usually designed for approximate or prespecified-sample inference.Continuous monitoring and data-dependent stopping require a sequentially valid construction.Source: §1 · p. 1
  • Fresh simulations from each candidate null normalize a positive predictable score into conditional e-values whose products form test martingales; inversion yields confidence sequences.Coverage holds for any positive predictable score and any number of reference observations.Source: §1 · p. 1
  • The likelihood ratio maximizes expected log growth within the normalized-score construction, motivating an amortized neural ratio estimator as a practical approximation.Expected log growth measures how quickly evidence accumulates against false candidates.Source: §1 · pp. 1–2
  • Neural scores yield confidence sequences close to conformalized likelihood ratios on synthetic benchmarks and comparable to application-specific scores in an LLM routing audit and microlensing inference.The applications use real model replies and real microlensing observations, respectively.Source: §1 · p. 2

2 RELATED WORK

SLICE combines simulation-based normalized-score e-values with sequential inference, distinguishing its parameter confidence sequences from prior fixed-sample methods and related hypothesis-testing work.

2 RELATED WORK

  • SLICE builds on simulation-based normalized-score e-values and sequential products, adding direct null simulation, predictable alternatives, and amortized neural ratio scores for parameter inference [47, 10].Its oracle-score and finite-reference results specialize earlier log-optimality and log-growth comparisons [10].Source: §2 · p. 2
  • Unlike ACORE, LF2I, WALDO, and TRUST approaches to fixed-sample inference, SLICE uses fresh candidate-null simulations and test-martingale inversion for time-uniform coverage [7, 8, 29, 3].Under the stated sampling conditions, validity holds for any positive predictable score and reference-batch size, without estimating critical values.Source: §2 · p. 2
  • The neural scores draw on classification-based amortized ratio estimation and contrastive learning, while normalization—not exact likelihood ratios or intrinsically valid neural scores—supplies validity.Score quality affects efficiency.Source: Neural ratio estimation · p. 2
  • Concurrent work studies likelihood-free anytime-valid hypothesis testing and related oracle bounds, whereas SLICE emphasizes parameter confidence sequences, adaptive alternatives, and empirical applications [14].The concurrent work additionally gives guarantees for approximate growth optimality of empirically trained scores [14].Source: Neural ratio estimation · p. 2

3 PRELIMINARIES

The preliminaries set the sequential sampling framework, explain how test martingales yield confidence sequences, and describe the neural ratio estimator used to approximate likelihood ratios.

3 PRELIMINARIES

  • The setting assumes i.i.d. observations from a fixed unknown parameter in a dominated model, with each new observation independent of information revealed earlier.The filtration includes observations and auxiliary randomness, including simulator draws.Source: §3 · p. 2

3.1 Test martingales and confidence sequences

  • Inverting a test supermartingale at threshold α^-1 gives a confidence sequence; the oracle likelihood-ratio martingale maximizes expected log growth when the divergence is finite.The oracle requires the unknown true parameter and unavailable likelihood ratios, so learned ratios cannot simply replace it while retaining the martingale guarantee.Source: §3.1 · p. 3

3.2 Neural ratio estimation

  • NRE-C trains a multiclass classifier to identify which candidate parameter generated an observation, with an additional independent-observation class; differences in network outputs estimate likelihood ratios.The marginal density cancels in the output differences.Source: §3.2 · p. 3

4 SEQUENTIAL UNCERTAINTY QUANTIFICATION FOR LIKELIHOOD-FREE MODELS

SLICE constructs likelihood-free test martingales by normalizing predictable scores with fresh null simulations, characterizes the score's efficiency, and implements the approach with neural ratio estimates and evaluation-grid inversion.

4.1 A likelihood-free test martingale

  • For each candidate null, conditional exchangeability of the observation and fresh reference simulations makes normalized positive predictable scores yield test martingales and time-uniform coverage.Reference batches may share randomness across candidates; conditional exchangeability is required under each candidate's own null, not independence across candidates.Source: §4.1 · pp. 3–4

4.2 The optimal score

  • Within the normalized-score construction, the likelihood-ratio score maximizes expected log growth against false candidates, while finite K limits the evidence obtainable per observation.The oracle score assigns the correct probability to each point being drawn from the alternative; expected log growth is bounded by log(K + 1).Source: §4.2 · pp. 4–5

4.3 Neural ratio scores

  • SLICE approximates the oracle score using a past-informed alternative and a neural likelihood-to-evidence network trained with NRE-C before monitoring.The network is kept fixed during monitoring, while the alternative is selected by minimizing accumulated evidence over the evaluation grid; other predictable estimates and network updates are permitted.Source: §4.3 · p. 5

4.4 Overall procedure

  • The overall procedure updates martingales across the evaluation grid and forms the output from that grid and the union of candidate confidence sets.Coverage beyond the evaluation grid requires additional control between grid points.Source: §4.4 · p. 5

5 EXPERIMENTS

The experiments assess coverage and efficiency on synthetic benchmarks, an LLM routing audit, and microlensing data. Neural-score sequences are close to likelihood-ratio or application-specific alternatives, while monitored fixed-sample intervals can exceed nominal miscoverage.

Experiments

  • The experiments compare sequential confidence-set coverage and efficiency across synthetic settings and applications, using α = 0.1 and finite candidate grids.Learned ratio estimators are fitted on independent data and held fixed during monitoring; widths and endpoints are averaged over repetitions with nonempty sets.Source: §5 · pp. 5–6

Synthetic experiments

  • In the Gaussian and g-and-k models, empirical miscoverage remains below α; conformalization and NRE-C cause little efficiency loss, while the oracle alternative gives narrower sets.Direct and conformalized likelihood-ratio sequences have nearly identical hull widths and candidate retention.Source: §5 · p. 6, Figure 1 · p. 7
  • Conformal NRE-C contains the true SLCP parameter in 94.6% of repetitions at n = 100, while retaining it and excluding a mirror image occurs in only 1.2%.Conformal NRE-C and conformal LR have similar candidate-exclusion rates; direct LR shrinks faster, and conformalization has a visible efficiency cost.Source: §5 · p. 6, Figure 2 · p. 7

Auditing model routing in an LLM API

  • Conformal count-ratio and NRE sequences keep miscoverage below 0.1 and have nearly identical, narrower hulls than Betting IRIS; fixed-time IRIS exceeds 0.1 under monitoring.The conformal sequences also detect substitution sooner than the fixed-time IRIS interval.Source: Figure 3 · p. 7, §5 · pp. 7–8

Inferring the initial mass function from microlensing events

  • In microlensing runs, conformal sequences keep miscoverage below α while the monitored Wilks interval exceeds it; NRE-C is almost as efficient as KDE.On real data, the θ1 sequences contain the published value −1.13 in 97% of orderings; the data say little against high θ2 values.Source: Figure 4 · p. 8, §5 · p. 8

6 DISCUSSION

SLICE separates confidence-sequence validity from score efficiency, and neural ratio scores provide a strong application-agnostic default. Its scope is bounded by candidate grids, reference-sampling randomness, alternative choice, and simulator specification.

Discussion

  • Normalizing any positive predictable score against fresh simulations yields exact likelihood-free confidence sequences; the score determines how quickly they shrink.Validity therefore does not require an accurate score.Source: §6 · p. 8
  • Neural ratio scores with a minimum-evidence alternative come close to exact likelihood ratios on synthetic benchmarks and application-specific scores in two real-data applications.The discussion presents them as an application-agnostic default.Source: §6 · p. 8
  • Efficiency is limited by grid-based candidates, randomness from reference sampling, the need for a good predictable alternative, and correct simulator specification.The discussion points to extensions for the first two limitations and prior information as a possible direction for the third.Source: §6 · p. 8

Sequential Likelihood-Free Inference via Conformal E-Values: Supplementary Materials

The supplement situates SLICE among simulation-based inference, calibration, sequential testing, and conformal methods. Its coverage argument applies the confidence-sequence definition at an arbitrary true parameter.

A ADDITIONAL RELATED WORK

  • Simulation-based inference includes posterior, density, and ratio surrogates, while related calibration methods address finite-budget posterior calibration and confidence regions [6, 22, 34].The cited CANVI guarantee is marginal under a simulator-induced joint distribution, unlike fixed-parameter or time-uniform coverage.Source: A ADDITIONAL RELATED WORK · p. 12
  • SLICE differs from calibrated confidence-region methods by forming conditional e-values at each observation and combining them into test martingales [7, 8, 29, 3].E-values, Ville’s inequality, and universal inference provide related foundations for sequential validity.Source: A ADDITIONAL RELATED WORK · p. 12
  • SLICE tests each candidate simulator distribution sequentially and inverts the tests, with fresh simulations as observed data arrive [10].This differs from anytime-valid Monte Carlo tests that add resamples for a fixed dataset.Source: A ADDITIONAL RELATED WORK · p. 12

B.1 Coverage of the confidence sequence

  • Applying the confidence-sequence definition at an arbitrary true parameter gives coverage of at least 1 − α at every time.The argument concludes that the probability the parameter remains in all confidence sets is at least 1 − α.Source: §B.1 · p. 13

B.2 Neural ratio estimation losses

The appendix characterizes neural ratio-estimation losses and their likelihood-ratio targets, then extends SLICE to conditional observations, projected targets, and averaged reference batches.

B.2 Neural ratio estimation losses

  • NRE-A’s population minimizer is the log likelihood-to-evidence ratio, with unequal class priors adding log{π/(1 − π)}.Source: §B.2 · p. 13
  • InfoNCE’s optimal score is log rθ1,θ0(y) up to a constant, and its classification task contrasts observations.Source: §B.2 · p. 13
  • NRE-B contrasts candidate parameters at fixed observations; its optimal scores are identifiable up to an observation-specific shift, while score differences recover log likelihood ratios.The population characterizations assume fθ(y)/m(y) is positive and finite almost everywhere; practical scores are restricted to the network family.Source: §B.2 · pp. 13–14
  • NRE-C adds a parameter-independent class, uniquely identifying log fθ(y)/m(y); NRE-A is recovered when Ktr = 1 and γ = 1.The added class removes the observation-specific ambiguity present in NRE-B.Source: §B.2 · p. 14

C.1.1 Beyond i.i.d. observations

  • Conditional validity extends to observations with covariates when fresh null simulations are conditionally independent and identically distributed given the past and current covariate.The covariate distribution may be unspecified; conditioning discards information about θ carried by covariates.Source: §C.1.1 · p. 15

C.1.2 Nuisance parameters and projections

  • Projecting a full confidence sequence through ψ preserves coverage but can be conservative; composite-null testing requires a separate construction because simple-null e-processes do not generally average validly.The resulting composite-null process gives an anytime-valid level-α test when it crosses α^-1.Source: §C.1.2 · p. 15

C.2.1 Averaging over independent reference batches

  • Averaging conditional e-values from B independent reference batches preserves the test-martingale property and divides conditional variance by B, at a cost of BK reference draws per observation and candidate.Source: §C.2.1 · p. 15

C.2.2 Grid evaluation and continuous parameter spaces

Continuous-parameter inversion requires certified interval bounds rather than ordinary finite-grid evaluation. The section gives procedures for obtaining and refining such bounds while preserving coverage.

C.2.2 Grid evaluation and continuous parameter spaces

  • Finite-grid evaluation guarantees coverage only for true parameters on the grid; continuous coverage requires a certified outer approximation that excludes intervals only when all contained parameters are certified rejected.Unresolved regions must be retained.Source: §C.2.2 · p. 16
  • Shared base draws define the test process simultaneously over Θ, and each fixed parameter retains the conditional-exchangeability validity argument.Source: §C.2.2 · p. 16
  • Adaptive interval subdivision and stopping require no multiplicity adjustment when unresolved regions remain included, because the numerical procedure only enlarges the ideal confidence sequence.Source: §C.2.2 · pp. 16, 18
  • Certified interval bounds can use piecewise-affine ReLU score behavior, but require verified activation partitions and same-time endpoint rejection; crossings at different times do not certify an interval.Unresolved boundary regions are retained.Source: §C.2.2 · pp. 17–18
  • Formal certification requires sound enclosures for rounding, activation boundaries, and threshold comparisons; learned-ratio accuracy does not affect coverage of the actual chosen score.Ordinary floating-point breakpoint calculations alone do not establish the required certificate.Source: §C.2.2 · p. 18

D IMPLEMENTATION AND COMPUTE

The implementation section reports computational costs and reproducibility details, then describes the simulators used in experiments and their likelihood properties.

D IMPLEMENTATION AND COMPUTE

  • Experiments use PyTorch and NumPy with a single base seed expanded into independent streams, running on an Apple Silicon M4 Pro with 24 GB memory.Source: D IMPLEMENTATION AND COMPUTE · p. 18
  • With G candidates and K references, each step uses GK simulator calls and (K + 1)G score evaluations, for O(nGK) total work and O(GK) memory over n steps.Training is a fixed pre-monitoring cost, and the confidence sequence is valid at any stopping time.Source: D IMPLEMENTATION AND COMPUTE · p. 18

E.1 Simulators

  • The Gaussian simulator has a closed-form likelihood, while the g-and-k benchmark lacks a closed-form inverse and requires numerical root solving for its density.Source: §E.1 · pp. 18–19
  • SLCP has a closed-form likelihood but four symmetric posterior modes because θ3 and θ4 enter only through their squares.Source: SLCP · p. 19

E.2 Training the neural ratio score

The neural-score diagnostics connect training quality to sequential efficiency: baseline scores achieve near-oracle log growth and confidence sequences, while an underparameterized NRE-A can lose localization power without invalidating coverage. The section also specifies the monitoring procedure and ablation comparisons.

E.2 Training the neural ratio score

  • The score network uses NRE-C with 20-way training batches, three width-128 residual blocks, and 10.24 million simulated pairs per model, then remains fixed during monitoring.Training uses streaming batches of 1,024 pairs; updates based only on past information during monitoring are permitted and affect efficiency, not validity.Source: §E.2 · p. 19

E.3 Running the confidence sequence

  • Confidence sequences use running intersections of candidates whose martingales have not crossed α^-1; predictable alternatives and reference simulations are based on past information or fresh draws.The SLCP analysis discards its first two e-values because its initialized alternative is far from θ∗; monitoring then begins at n = 3.Source: §E.3 · p. 20

E.4 Confidence-sequence diagnostics

  • Diagnostics assess cumulative miscoverage, hull width for location models, retained-candidate fraction for SLCP, and pair-level ratio accuracy alongside expected log growth.The small-network ablation compares exact and conformalized likelihood ratios, NRE-C, and NRE-A; it also varies reference count and predictable alternative.Source: §E.4 · p. 20, §F.2.1 · p. 23

F.1 Neural ratio quality (Gaussian and g-and-k)

  • At baseline capacity, neural confidence sequences are essentially indistinguishable from exact-likelihood-ratio sequences because both achieve near-oracle log growth.Across broad, local, and overall parameter pairs, both NRE objectives descend to the oracle growth line for Gaussian and g-and-k models.Source: §F.1 · p. 21
  • For baseline Gaussian and g-and-k models, both neural objectives and conformalized exact likelihood ratios have near-identical confidence-sequence widths and retention.Pair-level errors can grow off the diagonal without reducing efficiency, because expected log growth saturates for well-separated laws.Source: §F.1 · p. 21

F.1.1 Small architecture

  • With a two-unit network, NRE-A’s growth collapses on g-and-k, producing wide sets that barely localize θ∗, while NRE-C remains markedly more robust and coverage stays valid.NRE-A’s overall curve settles near −0.11 against an oracle near −0.81; its sets retain most candidates on g-and-k.Source: §F.1.1 · pp. 21, 23

F.2.2 Aggregated results

Across the ablations, conformal methods retain validity while their efficiency varies with reference count, alternative choice, and score capacity. The section also reports SLCP behavior and describes the semi-synthetic LLM-routing audit, its simulator, and score construction.

F.2.2 Aggregated results

  • Across baseline ablations, conformal neural and exact-likelihood-ratio procedures agree within Monte Carlo error; reducing references to K = 1 preserves validity but widens sets by roughly 20–40% at n = 100.For underparameterized NRE-A on g-and-k at θ∗ = 0 and 2, W100 = 5.5 on a width-8 grid reflects almost no localization.Source: §F.2.2 · p. 25, Table 2 · p. 26

F.2.3 Per-ablation confidence sequences (θ∗= 0)

  • At θ∗ = 0, baseline feasible procedures have superimposed trajectories, while truth-informed oracle alternatives give narrower sets; changing the alternative or shrinking the null budget affects width without breaching validity.Underparameterized NRE-A degrades mildly on Gaussian and severely on g-and-k, whereas NRE-C remains close to baseline.Source: §F.2.3 · p. 27, Figure 13 · p. 28

F.3 SLCP

  • For SLCP, cumulative miscoverage stays below α, and the three conformal sequences shrink similarly, with NRE-A slightly slower and direct likelihood ratio shrinking fastest.At n = 100, feasible procedures retain 4.1–8.2 candidates on average versus about four equivalent candidates for the oracles; neural scores need not preserve their symmetry.Source: §F.3 · p. 28, Table 4 · p. 30, Figure 17 · p. 32

G.1 Setting and data

  • The routing audit uses real replies from qwen3:8b and qwen3:1.7b in simulated routing streams; their replies barely overlap, so a single reply can strongly indicate which model answered.The study collected 28,000 replies per model and prompt; the audit estimates routing fraction θ from the share of atypical replies.Source: §G.1 · pp. 28–29

G.2 Simulator

  • The routing simulator models parsed replies as fθ(y) = (1 − θ)p(y) + θr(y), although the auditor cannot evaluate this likelihood because the models’ output pmfs are unavailable.The auditor can query the models to sample from their reply distributions.Source: §G.2 · p. 30

G.3 Computing the scores

  • The audit estimates the reply ratio from held-out model outputs using smoothed counts or binary NRE, then mixes scores with λ = 0.3 to improve set width without changing validity.On test streams, λ = 0.2 and 0.3 narrow final sets by about 30% relative to λ = 0 without slowing detection; λ = 0.5 can delay detection when routing is rare.Source: §G.3 · pp. 31–32

G.4 Experiments

The section reports the LLM routing audit and OGLE-IV microlensing experiments, including their simulation designs, confidence-sequence comparisons, and modeling scope. It also describes an exact reweighting approach for microlensing simulations and checks the fitted model against held-out data.

G.4 Experiments; G.5 Baselines

  • The audit runs 200 repetitions per prompt and routing fraction on 300-step streams, comparing conformal count-ratio and NRE sequences with an oracle count-ratio sequence and IRIS baselines.Betting IRIS extends the fixed-n IRIS test to a confidence sequence, but is only approximately valid because its tell rates are estimated.Source: §G.4 · p. 33, §G.5 · p. 34

G.6 Results

  • The conformal routing sequences stay below 0.1 cumulative miscoverage at every tested routing fraction, are narrower than Betting IRIS in most settings, and rule out honest service four to five times sooner at θ∗ = 0.1 and 0.3.At θ∗ = 0.3, they are narrower early on, after which Betting IRIS ends slightly narrower; the oracle alternative improves detection further.Source: §G.6 · p. 34, Table 7 · p. 35

G.7 Limitations

  • The routing simulator assumes independent queries routed only between the advertised model P and cheaper model R; other routing departures may have little power under its score.The score compares likelihood ratios within the mixture family {fθ}; detecting departures requires a score targeted to them.Source: §G.7 · pp. 34–35

H.1 Setting and data

  • For OGLE-IV, the analysis uses 1,798 odd-season events, while 1,716 even-season events are held out for model checking and selecting the semi-synthetic truth.The catalogue subset contains 3,514 events from 70 fields in the specified Galactic-coordinate region.Source: §H.1 · pp. 36–37

H.2 Simulator

  • The microlensing simulator models event timescales under a broken-power-law stellar initial mass function, with θ1 and θ2 as the low-mass-star and brown-dwarf slopes.Its likelihood is intractable because the timescale distribution integrates over distances, masses, and velocities.Source: §H.2 · pp. 37–39

H.3 Exact simulation by reweighting

  • To avoid rebuilding the Galaxy at each parameter, the authors reuse simulated event banks and accept entries with parameter-specific rejection probabilities to obtain exact draws.The approach uses a density ratio whose θ-dependent factors are computed exactly; the reported bank truncation bound is zero.Source: §H.3 · p. 39

H.4 Model checking

  • Held-out testing of the fitted microlensing model finds no evidence against it, with p = 0.09 using 20 bins and p = 0.19 using 40 bins.The fitted θ1 ≈ −1.1 agrees with the published −1.13, while θ2 is weakly identified, possibly because low-cadence fields detect few short events.Source: §H.4 · p. 40

H.5 Confidence sequences and baselines

  • For microlensing inference, the authors compare a conformal NRE-C score with a near-exact conformal KDE score and a fixed-n Wilks KDE interval.Semi-synthetic runs use θ∗ = (−1.1, 1.0) as truth; real-data runs process 200 random event orderings.Source: §H.5 · p. 40, Baselines · pp. 40–41

H.6 Results

In semi-synthetic microlensing runs, conformal NRE-C and KDE confidence sequences maintain the target miscoverage under continuous monitoring, unlike the fixed-n Wilks KDE interval. The NRE-C sequence is nearly as efficient as KDE for θ1, while the results indicate weak identification of θ2; on real data, both conformal sequences contain the published θ1 value in most orderings.

Semi-synthetic runs

  • At α = 0.1, conformal NRE-C and KDE confidence sequences keep empirical cumulative miscoverage below the threshold throughout, while the Wilks KDE CI has fixed-n miscoverage 0.09 at n = 1,798 but fails under continuous monitoring.The semi-synthetic experiment has 300 repetitions at θ∗ = (−1.1, 1.0).Source: §H.6 · p. 41, Table 9 · p. 41
  • For θ2 = 1.0, the NRE-score median lower endpoint is −0.95, while the upper endpoint exceeds 9 in 96% of repetitions with NRE and 97% with KDE.The sequences therefore exclude only low θ2 values after all 1,798 events, consistent with weak identification.Source: §H.6 · p. 41
  • For θ1, the NRE-C and near-exact KDE sequences end with nearly the same width, while even the oracle-alternative sequence remains slightly wider than the fixed-n Wilks KDE CI.Replacing the minimum-evidence alternative with the oracle substantially shrinks the sequence.Source: §H.6 · p. 41

Real-data runs

  • After 1,798 real events, NRE-C sets are nonempty in 99% of orderings and KDE sets in all; both contain published θ1 = −1.13 in 97% of orderings.They contain θ2 = −0.18 in 97.5% of orderings, and learn little about θ2 compared with the Wilks KDE CI.Source: §H.6 · p. 41
Loading 2610.13855v1…