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Estimating great expectations under autoregressive language models with potentials

Francesco I. Re, Shubhangi Ghosh, Tim Vieira, Ryan Cotterell

arXiv:2610.11399v1cs.AIcs.LGstat.ME

TL;DR

Estimating expectations of language-model test functionals can be costly when sampling-based estimators have high variance. The paper uses prefix potentials and next-token probabilities to construct unbiased estimators whose variance depends on the potential. Across several estimands and applications, it reports improved statistical efficiency at comparable computational cost.

  • Problem

    Expectations of test functionals under language models can be costly to estimate reliably because unbiased sampling estimators may have high variance.

    Source: §1 · p. 1

  • Method

    The paper uses real-valued prefix potentials to decompose test functionals into token-level increments and replaces increments with next-token conditional expectations in an unbiased estimator.

    Source: §1 · p. 2

  • Results

    Across four applications and several estimands, practical potentials improve statistical efficiency at comparable computational cost, with gains reported for losses, divergences, gradients, n-gram estimates, and event probabilities.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Potential design is a practical strategy for improving expectation-estimation efficiency across several language-model estimands and applications.

    Source: §5 · p. 9

    Practical potentials only approximate the zero-variance oracle and are not guaranteed to improve the baseline for every prompt; arbitrary potentials should not be assumed to yield similar gains.

    Source: LIMITATIONS · p. 15

Abstract

from arXiv · show

Many applications of language models hinge not on individual samples but on the expectation of a test functional under the model. Estimating such expectations reliably can be computationally expensive. In this paper, we show how to make estimation more efficient by exploiting the next-token conditional probabilities which are available as a by-product of sampling. We do so through potentials: real-valued functions on prefixes that decompose the test functional additively. We construct an estimator whose variance depends on the chosen potential, and derive conditions under which a potential reduces this variance. We then develop practical potentials for several estimands and applications, and demonstrate substantial variance reductions across several estimands at comparable computational cost.

1 INTRODUCTION

Many language-model applications require expectations of test functionals rather than individual text samples, but Monte Carlo estimation can be costly when variance is high. The paper introduces potentials that reuse next-token probabilities to reduce variance while preserving unbiasedness, and reports gains across several estimands and applications.

1 INTRODUCTION

  • Expectation estimation matters in applications including policy divergence, harmful-output probabilities, distillation, watermark detection, and interpretability.These examples estimate properties of the model’s distribution rather than evaluating an individual generated sample.Source: §1 · p. 1
  • Without exploitable structure, Monte Carlo approximations can require many samples because unbiased estimators may have high variance.The paper notes that dynamic programming can compute expectations when the model is a probabilistic finite-state automaton or probabilistic context-free grammar and the test function is additively decomposable.Source: §1 · p. 1
  • The framework decomposes a test functional into token-level increments and replaces each increment with its conditional expectation given the prefix, preserving unbiasedness while making variance depend on the decomposition.The next-token distributions used for this averaging are available as a by-product of sampling.Source: §1 · p. 2
  • Potentials are real-valued functions on prefixes that parameterize additive decompositions, redistributing function value across marginalizable token-level increments.The paper repurposes the potential construction from potential-based reward shaping in reinforcement learning.Source: §1 · p. 2
  • An oracle potential yields zero variance; practical design uses known additive structure or learns an approximation, while selection among candidates preserves unbiasedness using the full sample.The paper also derives an exact criterion for when an arbitrary potential reduces variance.Source: §1 · p. 2
  • Up to 17× effective sample size for losses and divergences, with variance reductions also reported for gradients, n-gram estimates, watermark detection, and importance-sampling estimates.For n-gram counts and frequencies, variance was about 99% lower than Monte Carlo after accounting for computational cost; watermark detection had a median 36% reduction, and the importance-sampling experiment a median 12% reduction.Source: §1 · p. 2

2 EXPECTATIONS UNDER AUTOREGRESSIVE LANGUAGE MODELS

The section defines autoregressive language models over strings and frames expectation estimation as a problem commonly requiring approximation. It reviews direct Monte Carlo and importance sampling, both unbiased but with variance that depends on the estimator or proposal.

2.1 LANGUAGE MODELS

  • A language model is a distribution over finite strings, with prefix probabilities inducing conditional next-token distributions and a left-to-right autoregressive factorization.The symbol set includes a distinguished terminal symbol EOS.Source: §2.1 · pp. 2–3

2.2 PROBLEM STATEMENT

  • The target is the expectation of a test function under the language model; exact computation is generally infeasible over the countably infinite string space.The section motivates approximation and notes that potential-based estimation can reuse sampled strings and retained next-token distributions.Source: §2.2 · p. 3

2.3 MONTE CARLO ESTIMATION

  • Monte Carlo estimates the expectation with an empirical average of independent samples; it is unbiased and strongly consistent when the expectation is finite.The strong law gives almost-sure convergence as the sample count grows.Source: §2.3 · p. 3

2.4 IMPORTANCE SAMPLING

  • Importance sampling rewrites the target expectation using proposal samples and likelihood weighting; it is unbiased under the stated support condition, but its variance may be above or below Monte Carlo’s.Variance is small when the weighted contribution varies little under the proposal; the proposal proportional to p(y)|f(y)| makes its magnitude constant.Source: §2.4 · p. 3

3 ESTIMATING EXPECTATIONS WITH POTENTIALS

The section turns prefix-level potentials into additive token contributions and uses next-token distributions to form a variance-reduction estimator. It establishes an oracle zero-variance potential and describes unbiased control-variate protection when practical choices are imperfect.

3.1 ADDITIVE DECOMPOSITIONS OVER PREFIXES: POTENTIALS

  • A potential assigns a real value to each prefix, and its successive-prefix increments express the test function as token-level contributions along a sampled string.At each prefix, the model’s next-token distribution permits averaging the next increment over possible tokens.Source: §3.1 · p. 4

3.2 THE POTENTIAL ESTIMATOR

  • The Bellman operator supplies expected next-token values, which are substituted into the sampled-prefix sum to form the potential estimator.The inner sum includes the final prefix, where EOS is drawn.Source: §3.2 · p. 4
  • When the sum of potential increments has finite second moment, the estimator is unbiased, and the oracle potential equals the conditional expectation of the test function given each prefix, yielding zero variance.The oracle is defined on prefixes with positive probability.Source: §3.2 · p. 4
  • The control-variate formulation gives variance equal to or below the unbiased baseline even when a chosen potential does not meet the variance-reduction condition.Estimating the coefficient or selecting among learned potentials from the same samples requires cross-fitting to preserve unbiasedness.Source: §3.2 · p. 5
  • For any fixed potential and coefficient, the control-variate estimator remains unbiased, with the coefficient chosen to minimize variance when the relevant second moments exist.Setting the coefficient to zero recovers the baseline estimator; setting it to one recovers the potential estimator.Source: §3.2 · p. 5

4 POTENTIAL DESIGN IN PRACTICE: APPLICATIONS AND EXPERIMENTS

The section develops analytical and learned potentials for several language-model estimands, then evaluates their variance and cost-adjusted efficiency across losses, n-gram statistics, and event probabilities. The experiments find substantial gains in many settings, with results varying by estimand and target.

4.1 ESTIMATING PROPER LOSSES, DIVERGENCES, AND GRADIENTS

  • For proper losses and their gradients, the observed-prefix function value gives an inexpensive analytical potential; the evaluation covers seven losses and divergences and their gradients.The token-level surprisals are computed during sampling, so this potential is free; for KL, the estimator coincides with Amini et al. (2025), whose claimed general variance ordering is challenged by the paper.Source: §4.1 · pp. 5–6

4.1.1 EXPERIMENTS AND RESULTS

  • 0.058–0.987 variance ratios across all 56 loss and divergence pair–quantity combinations show lower variance than Monte Carlo; effective sample-size gains reach 17.3×.Every bootstrap upper bound was below one.Source: Figure 1 · p. 6, §4.1.1 · p. 6
  • When gains were small for gradients, fitting the control-variate coefficient recovered substantial gains; variance reduction nevertheless did not hold for every prompt.Monte Carlo had lower variance on 1.5% of prompts overall, including 12.3% of T¨ulu-3 prompts; fitting α reduced the median ratio to Monte Carlo from 1.134 to 0.974 among KL prompts where the potential increased variance.Source: §4.1.1 · p. 6

4.2 ESTIMATING EXPECTED COUNTS AND FREQUENCIES OF N-GRAMS

  • For n-gram counts, the lookahead potential sums conditional probabilities that an occurrence starting at each position will be completed, while a terminal increment recovers the remainder.Marginalization costs O(LC) beyond generation, with a total per-target token-evaluation cost proxy of (L + 1)C; the experiment uses 7,647 targets and 4,000 GPT-2 samples per target.Source: §4.2 · p. 7, §4.2.1 · p. 7

4.2.1 EXPERIMENTS & RESULTS

  • 1.24% was the median cost-adjusted variance ratio for retained n-gram frequency targets, and 74.7% required at most 10% of Monte Carlo compute.Using upper confidence bounds, the median was 2.56%, with 95.6% still below parity; count gains were also largest for rare targets, while the simpler observed-prefix potential gave virtually no variance reduction.Source: §4.2.1 · pp. 7–8, Figure 2 · p. 8
  • At 4,000 samples, 79.7% of targets reached RSE ≤20% and 96.5% reached RSE ≤50%, versus 11.4% and 19.0% for Monte Carlo.For target lengths of at least four, Monte Carlo almost never reached RSE ≤50%, while the potential estimator did so for at least 90% in every length group.Source: Figure 2 · p. 8, §4.2.1 · p. 8

4.3 EVENT ANALYSIS WITH LANGUAGE MODELS

  • For event indicators, learned potentials approximate the oracle conditional event probability from prefix features; cross-fitting separates potential fitting from evaluation, and nested cross-fitting selects among candidates.The ARI experiment compares four candidates with MBAR and does not claim exact finite-sample unbiasedness for MBAR.Source: §4.3 · pp. 8–9, §4.3.1 · p. 9

4.3.1 EXPERIMENTS & RESULTS

  • 0.87 and 0.90 were the median variance ratios to MBAR for the mixture-model and logistic learned potentials, while nested selection attained 0.88.The terminal potential attained 0.92; the median 0.88 corresponds to an MBAR budget requiring 0.9 × 10^6 more generations than the potential estimator, assuming inverse scaling with sample size.Source: Table 3 · p. 9, §4.3.1 · p. 9

5 CONCLUSION

The paper presents potentials as a general framework for reducing variance in expectation estimates under autoregressive language models while preserving unbiasedness. Across several estimands and applications, practical potentials improve statistical efficiency at comparable computational cost.

5 CONCLUSION

  • The framework reuses next-token distributions available during sampling to reduce variance in expectation estimates under autoregressive language models.The estimator remains unbiased, while its variance depends on the chosen potential.Source: §5 · p. 9
  • Practical potentials improve statistical efficiency at comparable computational cost across several estimands and applications.The paper presents potential design as a general strategy for constructing lower-variance estimators under language models.Source: §5 · p. 9
  • The authors report using generative AI tools for research feedback, interpretation, mathematical checks, software development, and manuscript drafting and editing.They state that AI-assisted text and code were reviewed and verified by the authors, who take responsibility for the final work.Source: AI USE STATEMENT · p. 10

LIMITATIONS

The section qualifies the framework’s practical gains by identifying limits in potential quality, evaluation cost, and model access. It also relates the construction to prior estimation methods and records supporting results on importance sampling and unbiasedness.

LIMITATIONS

  • Practical potentials approximate the zero-variance oracle and may not improve every prompt; cross-fitting is needed when estimating a control-variate coefficient or selecting among learned potentials.A control-variate formulation can safeguard against poorly chosen potentials, but the studied gains do not imply that arbitrary potentials perform similarly.Source: LIMITATIONS · p. 15
  • Computational advantage depends on potential-evaluation cost: some quantities are available during sampling or cheap to compute, while elaborate potentials may require extra model evaluations or fitting.The authors identify inexpensive potentials with stronger variance reduction as a direction for future work.Source: LIMITATIONS · p. 15
  • The method requires next-token probabilities along sampled prefixes, so it applies directly when logits or token probabilities are exposed, not to output-only black-box APIs.Empirical conclusions are limited to the studied estimands, models, and configurations; gains may differ elsewhere.Source: LIMITATIONS · p. 15

A RELATED WORK

  • Prefix potentials connect to event-probability twists in importance sampling and to conditional Monte Carlo; the paper relates its oracle potential to optimal event twists.It distinguishes unbiasedness from variance reduction when discussing prior KL estimation, and describes its approach as complementary to Dorman et al. (2026).Source: A RELATED WORK · pp. 15–16

B.1 IMPORTANCE SAMPLING AND THE OPTIMAL PROPOSAL

  • For importance sampling, the variance-minimizing proposal is q(y) = p(y)|f(y)|/c, where c = E_p |f(Y)|; its minimum variance is zero only when f has constant sign under p.The proposal must cover every y for which p(y)|f(y)| is positive.Source: §B.1 · p. 16

B.2 UNBIASEDNESS OF THE POTENTIAL ESTIMATOR

  • The potential estimator is unbiased when its increments are absolutely summable in expectation.The proof uses conditioning on each sampled prefix and the telescoping identity for the potential increments.Source: §B.2 · pp. 16–17

B.3 PROOF OF PROPOSITION 1

The appendix establishes the oracle and control-variate properties, extends potential estimation to proposal sampling, and collects experimental details for watermark detection. For that application, it distinguishes an additive expected score from a non-additive detection event and describes a binomial approximation to the event’s oracle potential.

B.3 PROOF OF PROPOSITION 1

  • The oracle potential is the conditional continuation value; its expected increment vanishes at every positive-probability prefix, yielding a constant, zero-variance estimator.The proof obtains the result by conditioning on the next token and applying the one-step recursion.Source: §B.3 · p. 17

B.4 PROOF OF PROPOSITION 2

  • The control-variate estimator remains unbiased, and its optimal coefficient is Cov_p(bθ*, D)/Var_p(D) when Var_p(D) > 0.The coefficient minimizes a quadratic in the control-variate correction.Source: §B.4 · p. 17

B.5 POTENTIALS UNDER A PROPOSAL

  • Under proposal sampling, the importance-weighted potential estimator is unbiased, and its control-variate form at the optimal coefficient has variance no greater than the importance-sampling baseline.For θ ≠ 0, the oracle potential has zero variance if and only if q = p.Source: §B.5 · pp. 18–19

C ADDITIONAL ANALYSIS AND EXPERIMENTAL DETAILS

  • The appendix gathers the models and prompts, target-construction protocols, generation settings, and supporting figures and tables for the experiments.Its application subsections follow the order of the main-text experiments.Source: C ADDITIONAL ANALYSIS AND EXPERIMENTAL… · p. 19

C.1 ANALYSIS OF WATERMARK DETECTION

  • Watermark detection counts generated tokens in their prefix-dependent green lists and declares a string watermarked when the count exceeds a threshold.The setup follows Kirchenbauer et al. (2023), adding a logit bias to green-list tokens during generation.Source: §C.1 · p. 19

C.1.1 ESTIMANDS AND POTENTIAL DEFINITIONS

  • The expected watermark score is additive over prefixes, so the observed-prefix function is a potential; the non-additive detection indicator instead uses an approximation to the oracle conditional probability of eventual detection.The approximation uses a binomial-tail probability based on the green tokens still needed and a globally fitted green-token rate q for each length.Source: §C.1.1 · p. 19

C.1.2 EXPERIMENTS AND RESULTS

In watermark detection experiments, the binomial potential reduces variance for detection probability, especially when detection is uncertain, while the additive expected-score potential improves every tested prompt. The section also supplies potential definitions for loss and divergence estimands.

C.1.2 EXPERIMENTS AND RESULTS

  • 0.64 median variance ratio (36% reduction) across 14,366 non-deterministic prompt-configuration pairs; for 0.001 ≤ bπ < 0.5, the median ratio is 0.56 (44% reduction).The gain falls to a 0.97 median ratio for bπ ≥ 0.99; 92.7% of pairs improve, while the terminal potential remains close to Monte Carlo at 0.973 median ratio, without additional model evaluations.Source: Table 6 · p. 20, §C.1.2 · p. 20
  • Detected trajectories tend to pass through positive-potential regions, while undetected trajectories concentrate in low- or zero-potential regions.The figure illustrates how the potential contains information about the final detection outcome before generation ends.Source: §C.1.2 · pp. 20–21
  • The expected-score potential reduces variance for every prompt by 16–53%, with the largest gains for weak watermarks and short continuations.Across the twelve cells, the median ratio ranges from 0.471 at δ = 1, m = 32 to 0.837 at δ = 5, m = 200; the control-variate coefficient adds little.Source: §C.1.2 · p. 21

C.2.1 POTENTIAL DEFINITIONS

  • The appendix provides potential tables for losses and divergences, with gradient counterparts in another appendix; entropy uses increments −log p(a | x), and power-score regrets use β values from 1.25 to 2.As β approaches 1, the per-token power-score term divided by β − 1 tends to the KL term.Source: §C.2.1 · pp. 21–22

C.2.2 EXPERIMENTAL SETUP

The experiment spans eight policy–reference–horizon configurations across seven distinct model pairs, using a shared prompt-based sampling protocol. It includes sentiment-policy and chat-model comparisons.

C.2.2 EXPERIMENTAL SETUP

  • Eight configurations cover seven policy–reference pairs, with the GPT-2 sentiment pair evaluated at 50- and 256-token horizons.The 50-token setting reproduces Amini et al. (2025); the 256-token horizon matches the longer window used for the other models.Source: §C.2.2 · p. 22
  • Each configuration uses 512 prompts and B responses per prompt, with ancestral sampling at temperature 1 and no top-k or top-p filtering.Generation stops at EOS or the specified horizon, and sampling uses fp32 precision.Source: §C.2.2 · p. 22, Table 10 · p. 22
  • The GPT-2 sentiment policy is fine-tuned with DPO on 5,000 IMDb reviews, while the chat experiments compare Tülu-3 DPO with SFT and Gemma-2 SPPO iterations.The GPT-2 policy is evaluated on 512 IMDb test prompts with 4,000 responses each; chat models use 512 AlpacaEval 2 instructions and 100 responses each.Source: §C.2.2 · p. 22

C.2.3 ADDITIONAL RESULTS

The additional results examine how variance ratios relate to quantity magnitude and summarize prompt-level behavior for potential and cross-fitted control-variate estimators.

C.2.3 ADDITIONAL RESULTS

  • Figure 4 extends the magnitude–variance comparison from Figure 1(b) to all seven quantities, showing configuration-level aggregates alongside individual prompts.Source: §C.2.3 · p. 23
  • For KL and power-score regrets, variance reduction strengthens as policy–reference divergence decreases, especially for larger β; entropy and Brier loss stay closer to Monte Carlo.Figure 4 plots variance ratios against estimated quantity magnitude, with colored bank-level markers and gray prompt-level points.Source: Figure 4 · p. 23, §C.2.3 · p. 23
  • Table 12 compares per-prompt variance ratios for the potential and cross-fitted control-variate estimators, including prompts where the potential alone increases variance.Its rightmost columns report how often fitting the coefficient brings those ratios below one.Source: §C.2.3 · pp. 23–24, Table 12 · p. 24

C.2.4 REPRODUCING AMINI ET AL. (2025)

The appendix reproduces a KL-estimation result and develops gradient potentials with a shared vector-valued control-variate coefficient. The KL reproduction also shows that the unit coefficient is not always variance-optimal.

C.2.4 REPRODUCING AMINI ET AL. (2025)

  • At M = 1, fitting the control-variate coefficient lowers the standard deviation from 2.05 to 1.90 in the reproduced KL experiment.The experiment reproduces Amini et al. (2025) using the sentiment-tuned policy; the table reports means and empirical standard deviations across sample sizes.Source: §C.2.4 · p. 24, Table 13 · p. 24
  • A length-two-string counterexample shows the unit-coefficient estimator can remain unbiased while failing the variance-reduction condition when Monte Carlo is constant.The proposition-based comparison gives α⋆ = 0 in this case, whereas the estimator of Amini et al. (2025) corresponds to α = 1.Source: §C.2.4 · p. 24

C.3.1 POTENTIAL DEFINITIONS

  • For gradients, the estimator applies coordinatewise to a vector-valued test function, accounting for both changes in function value and sampling probability.For KL, the expected local increment is the KL divergence between next-token distributions, recovering the local gradient expression of Amini et al. (2025).Source: §C.3.1 · p. 25

C.3.2 CONTROL-VARIATE COEFFICIENT FOR VECTOR-VALUED FUNCTIONS

  • A single scalar coefficient is shared across all gradient coordinates and prompts, and is chosen to minimize total variance.Setting α = 0 recovers Monte Carlo and α = 1 recovers the potential estimator.Source: §C.3.2 · pp. 25–26

C.3.3 EXPERIMENTS AND RESULTS

Gradient potentials reduce variance across evaluated quantity–checkpoint comparisons, while the appendix also defines n-gram frequency estimation, its cost adjustment, and an observed-prefix potential.

C.3.3 EXPERIMENTS AND RESULTS

  • Variance ratios range from 0.11 to 0.98 across all 135 quantity–checkpoint comparisons; gains are generally largest early in DPO training.KL retains 35% of Monte Carlo variance at step 50 and 92% at step 938; β = 2 regret retains 11% and 65%, respectively. Fitting α⋆ further reduces entropy and Brier gradient ratios to 0.91 and 0.86.Source: §C.3.3 · p. 26, Figure 5 · p. 26

C.4.1 DEFINITION OF FREQUENCY ESTIMATOR

  • An n-gram frequency is expected target count divided by expected n-gram slots, estimated with a ratio whose finite-sample bias vanishes asymptotically under standard regularity conditions.The denominator is the Monte Carlo estimate of expected slots from the same generations.Source: §C.4.1 · pp. 26–27

C.4.2 DEFINITION OF COST-ADJUSTED VARIANCE RATIO

  • The cost-adjusted variance ratio estimates the fraction of Monte Carlo compute needed for equal variance, using a target-specific token-evaluation cost proxy.For frequency ratios, the comparison does not establish equal mean squared error because it excludes finite-sample bias.Source: §C.4.2 · p. 27

C.4.3 OBSERVED-PREFIX POTENTIAL

  • The observed-prefix potential marginalizes only at prefixes matching the substring through its final token; Table 16 groups GPT-2 unigram targets by occurrence count.The table reports median variance relative to Monte Carlo at equal sample size, M = 4,000.Source: §C.4.3 · p. 27, Table 16 · p. 27, Table 15 · p. 27

C.4.4 RESULTS FOR EXPECTED COUNTS

For expected counts, the potential estimator often reaches equal variance using a small fraction of Monte Carlo compute and yields more precise estimates at the same sample size. The remaining subsections report precision diagnostics and describe the ARI experiment, its potentials, selection procedure, and event-analysis results.

C.4.4 RESULTS FOR EXPECTED COUNTS

  • 0.79% of the Monte Carlo budget was the median cost-adjusted variance ratio for 4,149 retained targets; 95.4% remained below parity using interval upper bounds.Approximately 76.8% of retained targets required at most 10% of the Monte Carlo budget; the median upper bound was 1.78%.Source: §C.4.4 · p. 28, Figure 6 · p. 28
  • The cost ratio rose with expected count per document (Spearman rs = 0.87), and its median exceeded Monte Carlo parity for targets with one to ten expected occurrences per document.The median ratio rose from 2.3 × 10^-3 in the rarest populated decade to 1.24 for that more common group.Source: §C.4.4 · p. 28
  • At M = 4,000, 79.7% of targets reached RSE ≤20% and 96.5% reached RSE ≤50%, versus 11.4% and 19.0% for Monte Carlo.For targets with Lw ≥7, Monte Carlo observed none even in the 4 × 10^6-sample reference bank, while the potential estimator gave finite estimates for all.Source: §C.4.4 · p. 28, Figure 6 · p. 28

C.4.5 ADDITIONAL DIAGNOSTICS

  • Table 18 assesses whether targets remain below parity after accounting for estimation error, using conservative shares across cumulative relative-standard-error subsets.Its criterion requires the approximate normal interval for the cost-adjusted variance ratio to lie below parity.Source: §C.4.5 · p. 28, Table 18 · p. 30

C.5.1 THE AUTOMATED READABILITY INDEX

  • The ARI experiment estimates probabilities for 80 readability bins from TinyStories-8M completions of 100 tokens, using tilted and direct samples combined with MBAR weights.The experiment uses 19 tilt values, 200 chains of 32,400 retained states, and 200,000 direct samples.Source: §C.5.1 · p. 31

C.5.2 POTENTIAL DEFINITIONS

  • For each ARI bin, logistic potentials predict bin probabilities from prefix features, while mixture potentials forecast terminal ARI from extrapolated counts and sentence endings.The mixture potential assigns a Gaussian-mixture prediction's mass to the target bin.Source: §C.5.2 · p. 31

C.5.3 POTENTIAL SELECTION ALGORITHM

  • Nested cross-fitting selects a candidate by inner-fold variance, refits it on the outer training data, then evaluates it on the held-out fold.The described procedure uses five outer folds.Source: §C.5.3 · p. 32, Figure 9 · p. 32

C.5.4 COMPUTE COSTS AND ADDITIONAL RESULTS

  • Across all 64 populated ARI bins, the selected estimator’s median variance ratio was 0.88 relative to MBAR; among 38 reliable bins, the ratio showed no systematic dependence on baseline uncertainty.Single-candidate median variance reductions were 8% for the terminal potential and its calibrated version, 11% for logistic, and 13% for mixture, versus 12% for selection.Source: Figure 10 · p. 33, Figure 11 · p. 34, Figure 12 · p. 35

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