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arxiv: 2606.10096 · v1 · pith:F7MMNSHSnew · submitted 2026-06-08 · 📊 stat.ME

Estimating the Wasserstein barycenter of one-dimensional distributions under sparse sampling

James Peng , Florian Stijven , Linbo Wang , Peter B. Gilbert This is my paper

Pith reviewed 2026-06-27 15:15 UTC · model grok-4.3

classification 📊 stat.ME
keywords Wasserstein barycentersparse samplingdistributional dataquantile function estimationbinomial mixture modelsasymptotic normalityHIV sequence analysis
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The pith

The marginal-constructed barycenter estimator recovers the population mean of unit-level quantiles from sparse samples by estimating their distribution via marginal CDF mixtures.

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

This paper develops an estimator for the Wasserstein barycenter quantile function of one-dimensional distributional data when each unit is observed through only a small i.i.d. sample. The target at each quantile level is the mean of the corresponding latent unit-level quantiles. Instead of estimating full unit-level distributions, the method first recovers the distribution of those latent quantiles from the marginal distributions of the unit-level CDF values, which are estimated with binomial mixture models. The resulting marginal-constructed barycenter estimator is the mean of the recovered distribution. The authors establish its pointwise consistency and asymptotic normality and show in simulations that it outperforms the empirical barycenter under sparse sampling, with an application to HIV sequence data.

Core claim

The distribution of latent unit-level quantiles at a fixed level equals a functional of the marginal distributions of the unit-level CDF values at that level; these marginals are estimable by binomial mixture methods, so their functional yields a consistent, asymptotically normal estimator for the Wasserstein barycenter quantile that is less biased than the empirical version when per-unit samples are small.

What carries the argument

The marginal-constructed barycenter (MCB) estimator, formed by taking the mean of the distribution of latent unit-level quantiles recovered from binomial mixture estimates of the marginal unit-level CDF distributions.

If this is right

  • The MCB estimator is pointwise consistent for the population mean of the unit-level quantiles.
  • It is asymptotically normal under the conditions established in the paper.
  • It substantially reduces bias relative to the empirical Wasserstein barycenter when sampling per unit is sparse.
  • The estimator can summarize within-participant distributions of viral sequence features from the HVTN 502/503 trials when only a few sequences are available per participant.

Where Pith is reading between the lines

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

  • If the marginal representation extends, the same binomial-mixture step could be used to estimate other features of the distribution of quantiles, such as its spread.
  • The approach might apply to estimating other population functionals that involve averages of inverse CDFs under sparse per-unit observation.
  • Analogous marginal constructions could be sought for Wasserstein barycenters on non-real-line spaces.

Load-bearing premise

The distribution of latent unit-level quantiles at a given quantile level can be written in terms of the marginal distributions of the unit-level CDF values.

What would settle it

A dataset or simulation in which the per-unit sample size remains fixed while the number of units grows and the MCB estimator fails to converge to the true barycenter quantile despite accurate recovery of the marginal CDF distributions.

Figures

Figures reproduced from arXiv: 2606.10096 by Florian Stijven, James Peng, Linbo Wang, Peter B. Gilbert.

Figure 1
Figure 1. Figure 1: (a) The density of five bimodal distributions, (b) the Wasserstein barycenter of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantile functions of the empirical Wasserstein barycenter (red) and the proposed MCB [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the two-stage sampling scheme. Full arrows denote sampling, and dashed [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The figure shows the CDFs of five latent distributions. The vertical positions of the blue [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Empirical CDFs in a toy example with five unit-level distributions. The dotted [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results for Simulation Setting A. Rows display, from top to bottom, the mean estimate, [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results for Simulation Setting B. Rows display, from top to bottom, the mean estimate, [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimated barycenter quantile functions for the SLYNTVATL epitope distance and full [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
read the original abstract

We study distributional data under sparse sampling where each unit is represented by a probability distribution on the real line observed only through a small i.i.d.~sample. A natural notion of central tendency for one-dimensional distributional data is the Wasserstein barycenter, whose quantile function is the pointwise average of the unit-level quantile functions. We focus on pointwise estimation of the Wasserstein barycenter quantile function: at a given quantile level, the target is the population mean of the corresponding unit-level quantiles. A naive plug-in estimator is the empirical Wasserstein barycenter, which treats observed unit-level empirical distributions as the true latent unit-level distributions. Under sparse sampling, however, this estimator can be severely biased. We propose an approach that avoids directly estimating either the unit-level distributions or the full population law of distributions. We start with the more ambitious goal of characterizing the distribution of latent unit-level quantiles at a given quantile level. We show that this distribution can be written in terms of the marginal distributions of the unit-level CDF values, which can be estimated using binomial mixture methods. This motivates our estimator, the marginal-constructed barycenter (MCB) estimator, obtained by taking the mean of the estimated distribution of latent unit-level quantiles. We establish conditions under which the MCB estimator is pointwise consistent and asymptotically normal, and show through simulations that it can substantially outperform the empirical Wasserstein barycenter under sparse sampling. We illustrate the method in an analysis of HIV-1 sequence data from the HVTN 502/503 vaccine efficacy trials, using the barycenter to summarize and compare within-participant distributions of viral sequence features when only a small number of sequences are available per participant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the marginal-constructed barycenter (MCB) estimator for the quantile function of the Wasserstein barycenter of one-dimensional distributions observed under sparse sampling. It exploits the identity P(Q_i(p) ≤ x) = P(F_i(x) ≥ p) to express the distribution of latent unit-level quantiles in terms of marginal distributions of unit-level CDF values, which are estimated via binomial mixture methods; the MCB is then the mean of this estimated distribution. The authors state that they establish conditions for pointwise consistency and asymptotic normality of the MCB, demonstrate via simulations that it substantially outperforms the empirical Wasserstein barycenter under sparse sampling, and illustrate the method on HIV-1 sequence data from the HVTN 502/503 trials.

Significance. If the stated consistency and normality results hold under the supplied conditions, the MCB provides a bias-reduced estimator for distributional central tendency that avoids direct estimation of full unit-level distributions or the population law of distributions. This is a practical contribution for settings with limited per-unit samples. The approach receives credit for correctly leveraging a definitional identity without hidden dependence assumptions and for including both simulation comparisons and a real-data application.

major comments (2)
  1. [asymptotic results section] The section establishing asymptotic normality: the explicit regularity conditions on the binomial mixture estimators (including rates for estimating the marginal CDF distributions and any requirements on the number of samples per unit or support assumptions) must be stated with sufficient precision to verify the claimed asymptotic normality; without these, the support for the central theoretical claim remains incomplete.
  2. [consistency theorem] The consistency theorem: the paper asserts pointwise consistency under certain conditions, but the load-bearing step of propagating the mixture estimation error to the mean of the induced quantile distribution requires an explicit error bound or convergence argument that is not verifiable from the high-level description.
minor comments (2)
  1. [simulations] The simulation section should report the exact distributions, sample sizes per unit, and number of Monte Carlo replications to allow direct reproduction of the reported outperformance.
  2. [methodology] Notation for the binomial mixture parameters should be standardized and cross-referenced to the estimation procedure to improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and agree that the theoretical sections require greater precision in the stated conditions and arguments.

read point-by-point responses

  1. Referee: [asymptotic results section] The section establishing asymptotic normality: the explicit regularity conditions on the binomial mixture estimators (including rates for estimating the marginal CDF distributions and any requirements on the number of samples per unit or support assumptions) must be stated with sufficient precision to verify the claimed asymptotic normality; without these, the support for the central theoretical claim remains incomplete.

    Authors: We agree that the regularity conditions on the binomial mixture estimators must be stated with greater precision, including explicit rates for marginal CDF estimation, requirements on per-unit sample sizes, and support assumptions, to rigorously support the asymptotic normality claim. In the revised manuscript we will expand the asymptotic results section with a dedicated subsection listing these conditions and the associated convergence rates that close the argument. revision: yes

  2. Referee: [consistency theorem] The consistency theorem: the paper asserts pointwise consistency under certain conditions, but the load-bearing step of propagating the mixture estimation error to the mean of the induced quantile distribution requires an explicit error bound or convergence argument that is not verifiable from the high-level description.

    Authors: We acknowledge that the consistency proof requires an explicit error bound showing how the mixture estimation error propagates to the mean of the induced quantile distribution. In the revision we will strengthen the consistency theorem by inserting a detailed convergence argument that derives the required bound from the uniform consistency of the binomial mixture estimators. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction uses the identity P(Q_i(p) ≤ x) = P(F_i(x) ≥ p), which is a direct consequence of the definitions of the quantile and CDF functions and holds without reference to the target estimator or any fitted quantities. Marginal distributions of the F_i(x) are recovered via binomial mixture methods applied to independent observations, after which the mean of the induced distribution of Q_i(p) is taken to form the MCB estimator. Consistency and asymptotic normality are stated as separate results under explicit conditions. No step reduces a claimed prediction to a fitted input by construction, no self-citation is load-bearing for the core identity or estimator, and the derivation remains self-contained against external statistical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the core modeling step is the representation of the quantile distribution via marginal CDFs, and binomial mixture estimation likely involves fitted parameters whose details are not provided.

free parameters (1)
  • mixture parameters in binomial model
    Binomial mixture methods for estimating marginal CDF distributions typically require fitting mixture weights or components to data.
axioms (1)
  • domain assumption The distribution of latent unit-level quantiles can be written in terms of the marginal distributions of the unit-level CDF values
    This representation is the key step that motivates the estimator and is stated directly in the abstract.

pith-pipeline@v0.9.1-grok · 5846 in / 1245 out tokens · 29992 ms · 2026-06-27T15:15:02.608764+00:00 · methodology

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