Partial pooling predicts cross-validation reliability: a closed-form triage and Rao-Blackwellised cure for hierarchical LOO

Aidan D Bindoff

REVIEW 3 major objections 4 minor 18 references

In hierarchical models, the observations where cheap leave-one-out cross-validation collapses can be predicted in advance from the pooling factor, and a marginalised estimator corrects them without refits.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-01 14:10 UTC pith:UDT2WQCL

load-bearing objection A genuinely useful and unusually honest methods paper; the core triage works in the tested regimes, and the RB-LOO cure is right, but the 'refit only flagged folds' guarantee is narrower than the abstract suggests. the 3 major comments →

arxiv 2607.18836 v1 pith:UDT2WQCL submitted 2026-07-21 stat.ME stat.CO

Partial pooling predicts cross-validation reliability: a closed-form triage and Rao-Blackwellised cure for hierarchical LOO

Aidan D Bindoff This is my paper

classification stat.ME stat.CO MSC 62F1562G0962J12

keywords cross-validationleave-one-outPSIS-LOOhierarchical modelspartial poolingimportance samplingpooling factorstructural leverage

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to show that the standard cheap approximation to leave-one-out cross-validation (PSIS-LOO) fails in hierarchical models in a predictable, structured way: exactly on the folds where a random-effect coordinate is driven by the data and its group is small. It argues that the pooling factor—the share of a group's posterior precision that comes from the prior—and an associated per-observation structural leverage identify these failing folds without forming any importance weights, and in Gaussian linear mixed models the map is determined by group sizes alone before outcomes are seen. The proposed cure, RB-LOO, marginalises the random-effect block and importance-samples only the fixed-effect and variance parameters, in closed form for Gaussian blocks and one-dimensional quadrature for Bernoulli, binomial and Poisson blocks. If the paper is right, the most troublesome failures of routine cross-validation can be flagged in advance and corrected with zero refits, and model-comparison verdicts that look decisive under the standard tool can be shown to be an artefact of the approximation.

Core claim

The paper's core claim: in hierarchical models, the folds where PSIS-LOO fails are exactly those where deleting one case moves a data-driven random-effect coordinate, and the pattern is visible before any importance weights. The pooling factor and structural leverage separate failing folds (khat>0.7) with AUC 0.96 in Gaussian LMMs from group sizes alone, and AUC 0.81 in logistic GLMMs from one fit. RB-LOO marginalises the random-effect block and importance-samples only base parameters (analytic Gaussian downdate; 1-D quadrature for Bernoulli/binomial/Poisson). On 97 failing overdispersion folds it matches exact refits (elpd RMSE 0.041 vs 0.638) and flips a model-selection z from 4.9 to 1.0;

What carries the argument

The central object is a geometric split of each observation's case-deletion influence, obtained as a Schur complement of the Fisher information: a vertical term living in the random-effect direction, which sums to the pooling factor and predicts where PSIS-LOO fails, and a horizontal term living in the base (fixed-effect/variance) direction, which predicts where the replacement estimator is itself strained. The estimator that carries the cure, RB-LOO, is a variance-reduced importance-sampling scheme: instead of reweighting the random effect, it marginalises it out of the leave-one-out predictive—analytically via a rank-1 Gaussian downdate, or by one-dimensional quadrature for scalar Bernoull

Load-bearing premise

The load-bearing premise is that deleting one observation leaves the posterior of the fixed effects and variance components essentially unchanged, so that reweighting the full-data posterior over only those base parameters is safe; when a single deletion moves the base posterior appreciably—few groups, weak identification—RB-LOO itself needs an exact refit.

What would settle it

Run RB-LOO and exact leave-one-out refits on a Gaussian LMM with only 4–6 groups and singleton observations; if the residual base-khat diagnostic fails to separate the folds where RB-LOO's elpd deviates from exact refits by more than 0.25 nats (the paper reports AUC 0.93), the two-level triage claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Model-selection verdicts in hierarchical and overdispersed data can be artifacts of the approximation: the paper shows a decisive-looking z=4.9 difference becoming z=1.0 under exact refits, with RB-LOO reproducing the exact refit.
In Gaussian LMMs the failure map is available before outcomes are collected, so group sizes alone can warn which observations will need special handling.
The drop-in implementation lets a standard workflow flag the risky folds, marginalise them, and refit only the residual few percent, recovering refit-quality LOO at a small fraction of the cost.
On singleton-driven failures RB-LOO is substantially more accurate than the standard moment-matching repair—about 3x pooled on logistic GLMMs—and corrects failures that moment matching leaves in place on real count data.
The method applies to single random-intercept groupings; crossed or multiple grouping factors are explicitly out of scope and fall back to plain PSIS-LOO.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the pooling-factor-to-reliability map could be used at study design time in hierarchical settings: given planned group sizes and a prior guess at the variance component, one could flag in advance the observations whose deletion would make cross-validation unstable.
Beyond the paper, the decision flip from a per-observation latent model versus a marginal alternative suggests that latent-vs-marginal model comparisons may carry systematic PSIS-LOO bias that does not cancel in the difference; testing this on other latent structures (mixtures, factor models) is a natural extension.
Beyond the paper, the base-fiber split is a geometric claim that likely applies to other case-deletion diagnostics, not just PSIS-LOO; it gives a template for predicting when any leave-one-out approximation that reweights the full posterior will have heavy-tailed weights.
Beyond the paper, combining the residual base-khat flag with an automated exact-refit fallback would produce a fully adaptive workflow that never silently trusts the approximation; the paper reports the flag's accuracy but leaves the automation as a design choice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

A genuinely useful and unusually honest methods paper; the core triage works in the tested regimes, and the RB-LOO cure is right, but the 'refit only flagged folds' guarantee is narrower than the abstract suggests.

read the letter

Short version: this paper is worth engaging with. The new thing is the pooling-factor-to-PSIS-reliability map (C1) and the packaged observation-level closed forms for RB-LOO, plus the head-to-head showing marginalisation beats moment matching on singleton-heavy logistic GLMMs. The marginalisation itself is prior art, and the paper says so clearly. What impressed me is the honesty infrastructure: exact refits as gold standard, per-replicate MCSE, explicit statements about what the experiments can and cannot falsify, and a limitations section that names the estimand-sharing issue on singleton folds. That is how you do simulation work.

The core empirical claims hold up in the tested regimes. On the overdispersed epilepsy data, RB-LOO reproduces an 82-minute exact refit with RMSE 0.04 while PSIS-LOO and moment matching are over-optimistic and even biased on sub-threshold folds. The model-selection flip (z=4.9 to z=1.0) is dramatic but the paper backs it with exact refits, so it's credible.

Soft spots, in proportion. The two-level triage (C3) is the least secured claim. The 'refit only 3% of folds and still match exact LOO' result comes from a single regime: few-groups Gaussian LMMs with weak base identifiability. The residual base-khat flag is summarized only by AUC 0.93, not by a confusion matrix at the 0.7 threshold. AUC is rank-based; it doesn't tell you how many diverging folds get sent to refit or missed. With only 65 flagged folds among 2340, a missed divergence could shift the aggregate RMSE. The stress-test note is right about this. It's not a fatal flaw—the combined RMSE 0.067 is reported directly—but the operating point deserves a table, and the experiment should be extended to logistic/Poisson blocks with small J before calling it a general guarantee.

Also the 'design-time' framing is a bit strong even for Gaussian LMMs: the leverage ranking depends on the fitted variance ratio, so it's post-fit unless you treat the ratio as known. The paper's own honesty section acknowledges this, but the abstract oversells it.

Who gets value: anyone doing hierarchical model comparison with loo, and methodologists working on PSIS repairs. It deserves a serious referee. I'd send it out, with a request to add threshold-level diagnostics and broaden the few-groups stress test.

Referee Report

3 major / 4 minor

Summary. The paper studies why PSIS-LOO fails for hierarchical models and proposes a two-part remedy: a weight-free triage based on the Gelman–Pardoe pooling factor and the associated structural leverage (C1), and an integrated importance-sampling estimator, RB-LOO, that marginalises the random-effect block before forming base-level importance weights (C2). A Schur-complement decomposition (C3) separates case-deletion influence into a vertical/pooling component, claimed to govern PSIS-LOO failures, and a horizontal/variance-component component, claimed to govern residual RB-LOO failures. The empirical sections validate the predictor against exact refits in Gaussian LMMs and logistic GLMMs, benchmark RB-LOO against moment matching on singleton-heavy designs, and use the overdispersed epilepsy data to show that RB-LOO changes a model-selection verdict. The paper is explicit about the prior art for marginalisation, about pooled AUC optimism, and about the limited regime in which RB-LOO itself is stressed.

Significance. If the main claims hold, the paper provides a practical and conceptually useful advance: a principled, closed-form predictor of where PSIS-LOO will fail, and a cheap marginalisation-based repair that is well matched to the failure mode. The algebraic propositions are clean and the reproduction scripts are a genuine strength. The paper also makes honest, explicit limitation statements that help bound the claims. The least secured part is the two-level triage claim in C3: the operational threshold for sending folds to exact refit is validated only by a rank-based AUC, not by threshold-level sensitivity/specificity, and the stress-test design covers only Gaussian LMMs with few groups. Because the abstract's 'refitting only the few folds that need it' guarantee rests on this operating point, this needs additional evidence or a more carefully qualified statement.

major comments (3)

[Section 5.5 (E-basestress) and Section 4 (refit flag)] The protocol in C3 refits folds with residual base-khat > 0.7, yet the supporting evidence is summarized only by pooled AUC 0.93 for separating |Delta(elpd)| > 0.25 divergences. AUC is rank-based and does not characterize sensitivity/specificity at the 0.7 cutoff. With 65 flagged folds among 2340, the paper should report the confusion matrix at the threshold, the missed-divergence count and magnitude, and the hybrid RMSE under a range of thresholds. The statement that the two-level triage 'recovers the exact answer' is an operating-point claim, and the current reporting does not establish it.
[Section 5.5 and Abstract (generality of C3)] The stress test of RB-LOO is limited to Gaussian LMMs with J=4-6 groups, singletons, and a weakly identified sigma_u; no variation in likelihood family, base dimension, or group structure is exercised. Since the abstract generalizes to 'the few folds that need it' and Section 5.4 uses the result to support a decision-changing claim, the authors should either add stress-test configurations covering broader hierarchical settings (e.g., logistic GLMMs, more base parameters) or explicitly restrict the C3 guarantee to the demonstrated regime. This is load-bearing for the exact-recovery claim.
[Section 4 and Abstract ('design-time' wording)] In the Gaussian LMM case, the leverage reduces to group size plus a fitted scalar sigma_u, so the map is available after a variance-component estimate, not literally before seeing outcomes. The paper's Section 7 honesty note is accurate, but the abstract and Section 4 use 'a-priori' and 'design-time' in ways that could mislead readers. Please carry the Section 7 qualification into the abstract and introduction.

minor comments (4)

[Section 5.5] The text reports only AUC for the residual base-khat flag. A calibration plot or a table of sensitivity/specificity for khat thresholds near 0.7 would greatly improve the practical usability of the triage.
[Reproducibility section] 'rbloopackage' appears without a space/backtick in the code/software paragraph; check formatting. Consider adding a versioned release URL for the package.
[Section 2] The statement that the marginalised estimator is 'the standing recommendation of the loo documentation' is helpful, but citing the specific documentation version would make the provenance checkable.
[General] The paper's own limitation notes are commendable. In particular, the acknowledgment that pooled AUCs are optimistic and that per-replicate Spearman is primary is exactly the right way to report dependent folds; the same standard should be applied to the AUC-0.93 threshold analysis in Section 5.5.

Circularity Check

0 steps flagged

No circular derivation chain: C1 is a design/post-fit diagnostic not fitted to the khat outcomes, RB-LOO is checked against exact refits, and the sole self-citation (Bindoff 2026) supplies geometry and notation rather than load-bearing content.

full rationale

The claimed derivation chain is not circular. C1 predicts PSIS-LOO failures from the Gelman-Pardoe pooling factor and structural leverage; these quantities are computed from the Fisher blocks and (for GLMMs) one fitted variance component, with the paper stating they are 'formed without importance weights and without case deletion'. There is no parameter fitted to the khat>0.7 outcomes, so the reported AUC values are empirical associations, not identities. The RB-LOO estimator is integrated importance-sampling LOO, which the paper explicitly credits to Merkle et al., Vehtari et al., Bürkner et al. and the loo documentation; calling it Rao-Blackwellised is disclosed as naming a standard operation ('integrated IS is Rao-Blackwellisation'), not a new derivation. Its accuracy is measured against exact refits / reloo, and the paper is transparent that on singleton folds it and reloo compute the same marginal integral, so those comparisons cannot falsify it; the independent falsification is Section 5.5 (E-basestress), where RB-LOO errs on flagged folds and matches on unflagged folds. The C3 refit flag uses residual base-khat >0.7, the standard PSIS diagnostic applied to the base weights, and is validated by exact-refit RMSE; there is no construction by which the flag is defined to equal the RB-LOO error. The only self-citation, Bindoff (2026), supplies base-fiber coordinates and the identity sum_i h_i = 1 - pi_j; Proposition 1 is proved in the paper by a Schur complement, and the vertical-term equality is called an exact consequence of the Gelman-Pardoe definition, so the prior work is interpretive/nomenclatural, not load-bearing. No uniqueness theorem from the authors is invoked. Because the central predictions are not fitted to their targets and the estimator is validated against independent refits, there is no circular step; score 1 reflects only the minor, non-load-bearing self-citation in the geometric framing.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central claims rely on no free parameters fitted to the target; the quadrature grid is a hand-chosen approximation parameter, and the standard khat > 0.7 threshold comes from Vehtari et al. The axioms are standard matrix inversion, Rao-Blackwell, and domain assumptions about base-stability and single random-intercept scope.

free parameters (1)

Quadrature grid (nodes, range) = 64 nodes, ±6σ_u; widened for extreme singleton responses
Implementation parameter for 1-D numerical integration of the random-effect block. Robustness demonstrated by grid-independence (stable to 0.1 nat), so it does not affect the central claims.

axioms (5)

standard math The Fisher information block G is positive definite and G_FF is invertible, so the Schur decomposition of Prop. 1 is valid.
Standard regularity condition for the block-inverse identity; satisfied for identified mixed models.
standard math Rao-Blackwell inequality (Prop. 2) applies: E[W_i|θ] is unbiased and has no greater variance than W_i.
Classical conditional-expectation inequality; the paper notes that finiteness of Var(W-bar_i) is an additional assumption on base identifiability, not implied by the inequality.
domain assumption The single full-data fit's Fisher information and variance components, evaluated at the posterior mean for GLMMs, are informative about leave-one-out importance-sampling behaviour.
C1 triage is post-fit for GLMMs; the paper validates this empirically (AUC 0.81–0.96).
domain assumption The base posterior (fixed effects and variance components) is stable under single-case deletion, except on folds flagged by the residual base-khat.
RB-LOO's residual importance sampling relies on this; Section 5.5 shows it fails on flagged folds in few-groups designs, where a refit is required.
domain assumption The random-effect block is a single scalar random intercept per group; crossed or multiple grouping factors are out of scope.
Explicitly declared scope of the method; quadrature becomes multi-dimensional for slopes/correlated effects, and the package falls back to PSIS for crossed factors.

pith-pipeline@v1.3.0-alltime-deepseek · 12546 in / 16657 out tokens · 147245 ms · 2026-08-01T14:10:28.709967+00:00 · methodology

0 comments

read the original abstract

For hierarchical models, Pareto-smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) fails on the folds where a random-effect coordinate is data-driven and its group is small. We show that the Gelman-Pardoe pooling factor and structural leverage predict these folds from model structure and group sizes, without forming importance weights. In Gaussian linear mixed models the leverage reduces to group size, giving a design-time map that separates the failing ($\hat{k}>0.7$) folds with AUC 0.96; across replicated logistic GLMMs the post-fit, weight-free predictor reaches AUC 0.81. The cure is integrated importance sampling: marginalise the random-effect block and importance-sample only the base parameters. This is not new, but we contribute its observation-level specialisation for random-intercept GLMMs: an analytic Gaussian downdate and a 1-D quadrature for Bernoulli, binomial and Poisson responses, packaged as a drop-in rb_loo(fit). Against exact refits, this marginalised estimator (RB-LOO) is $3\times$ more accurate than moment matching on singleton-heavy logistic GLMMs. On overdispersed count data with 97 failing folds, moment matching leaves 37 uncorrected and is no more accurate than raw PSIS-LOO, while RB-LOO reproduces the 82-minute exact refit (elpd RMSE 0.04) at no cost. The error changes decisions: against a negative-binomial model, PSIS-LOO reports decisive evidence ($z=4.9$) and reloo reports significant evidence ($z=3.4$) for the more complex model, where an exact analysis, reproduced by RB-LOO, finds the two indistinguishable ($z=1.0$). A base-fiber Schur decomposition splits case-deletion influence into a vertical (pooling) term that governs where PSIS-LOO fails and a horizontal (variance-component) term that governs where RB-LOO is itself strained, giving a two-level triage that recovers the exact answer while refitting only the few folds that need it.

Figures

Figures reproduced from arXiv: 2607.18836 by Aidan D Bindoff.

**Figure 2.** Figure 2: Moment-matching head-to-head (E-MM), logistic GLMM. On the high- [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Epilepsy, observation-level overdispersion model. Left: structural leverage predicts PSIS [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Model selection under three estimators (E-decision), observation-level Poisson against [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: RB-LOO’s own limit and the refit flag (E-basestress), few-groups Gaussian LMMs. Left: [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Efficiency in the number of draws (E-Sscaling). On the high- [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 6 canonical work pages · 1 internal anchor

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Bindoff, A. D. (2026). A flat connection: the pooling factor and the geometry of centring in hierarchical MCMC. arXiv:2606.19655. doi:10.48550/arXiv.2606.19655

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.19655 2026
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Bürkner, P.-C., Gabry, J., and Vehtari, A. (2021). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student- t models. Computational Statistics, 36:1243--1261. doi:10.1007/s00180-020-01045-4

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Pith/arXiv arXiv 2010
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C., Furr, D., and Rabe-Hesketh, S

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[1] [1]

Bindoff, A. D. (2026). A flat connection: the pooling factor and the geometry of centring in hierarchical MCMC. arXiv:2606.19655. doi:10.48550/arXiv.2606.19655

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.19655 2026

[2] [2]

Bürkner, P.-C., Gabry, J., and Vehtari, A. (2021). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student- t models. Computational Statistics, 36:1243--1261. doi:10.1007/s00180-020-01045-4

work page doi:10.1007/s00180-020-01045-4 2021

[3] [3]

Held, L., Schrödle, B., and Rue, H. (2010). Posterior and cross-validatory predictive checks: a comparison of MCMC and INLA. In Statistical Modelling and Regression Structures, 91--110. Physica-Verlag. doi:10.1007/978-3-7908-2413-1\_6

work page doi:10.1007/978-3-7908-2413-1 2010

[4] [4]

S., Kong, X., and Carlin, B

Cui, Y., Hodges, J. S., Kong, X., and Carlin, B. P. (2010). Partitioning degrees of freedom in hierarchical and other richly parameterized models. Technometrics, 52(1):124--136. doi:10.1198/TECH.2009.08161

Pith/arXiv arXiv 2010

[5] [5]

and Pardoe, I

Gelman, A. and Pardoe, I. (2006). Bayesian measures of explained variance and pooling in multilevel (hierarchical) models. Technometrics, 48(2):241--251. doi:10.1198/004017005000000517

doi:10.1198/004017005000000517 2006

[6] [6]

Hodges, J. S. and Sargent, D. J. (2001). Counting degrees of freedom in hierarchical and other richly parameterized models. Biometrika, 88(2):367--379. doi:10.1093/biomet/88.2.367

work page doi:10.1093/biomet/88.2.367 2001

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Liu, Z., Van Niekerk, J., and Rue, H. (2022). Leave-group-out cross-validation for latent Gaussian models. arXiv:2210.04482

Pith/arXiv arXiv 2022

[8] [8]

Lovison, G., Sciandra, M., Albano, A., and Di Maria, C. (2026). The augmented hat-matrix of hierarchical generalised linear models and its use in leverage diagnostics. International Statistical Review. doi:10.1111/insr.70030

work page doi:10.1111/insr.70030 2026

[9] [9]

C., Furr, D., and Rabe-Hesketh, S

Merkle, E. C., Furr, D., and Rabe-Hesketh, S. (2019). Bayesian comparison of latent variable models: conditional versus marginal likelihoods. Psychometrika, 84(3):802--829. doi:10.1007/s11336-019-09679-0

work page doi:10.1007/s11336-019-09679-0 2019

[10] [10]

Paananen, T., Piironen, J., Bürkner, P.-C., and Vehtari, A. (2021). Implicitly adaptive importance sampling. Statistics and Computing, 31:16. doi:10.1007/s11222-020-09982-2

doi:10.1007/s11222-020-09982-2 2021

[11] [11]

Peruggia, M. (1997). On the variability of case-deletion importance sampling weights in the Bayesian linear model. Journal of the American Statistical Association, 92(437):199--207. doi:10.1080/01621459.1997.10473617

arXiv 1997

[12] [12]

Rue, H., Martino, S., and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society B, 71(2):319--392. doi:10.1111/j.1467-9868.2008.00700.x

arXiv 2009

[13] [13]

Sivula, T., Magnusson, M., and Vehtari, A. (2020). Uncertainty in Bayesian leave-one-out cross-validation based model comparison. arXiv:2008.10296

arXiv 2020

[14] [14]

Silva, L. A. and Zanella, G. (2024). Robust leave-one-out cross-validation for high-dimensional Bayesian models. Journal of the American Statistical Association, 119(547):2369--2381. doi:10.1080/01621459.2023.2257893

arXiv 2024

[15] [15]

Vehtari, A., Mononen, T., Tolvanen, V., Sivula, T., and Winther, O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. Journal of Machine Learning Research, 17(103):1--38

2016

[16] [16]

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5):1413--1432. doi:10.1007/s11222-016-9696-4

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