On the relationship between the Wasserstein distance and differences in life expectancy at birth

Markus Sauerberg

arxiv: 2508.17235 · v6 · submitted 2025-08-24 · 📊 stat.AP

On the relationship between the Wasserstein distance and differences in life expectancy at birth

Markus Sauerberg This is my paper

Pith reviewed 2026-05-18 21:56 UTC · model grok-4.3

classification 📊 stat.AP

keywords Wasserstein distancelife expectancy at birthage-at-death distributionsurvivorship functionmortality comparisonoptimal transport

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The pith

The Wasserstein distance between two age-at-death distributions equals the gap in life expectancy at birth when survivorship functions do not cross.

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

This paper shows that the Wasserstein distance between age-at-death distributions equals the absolute difference in life expectancies at birth whenever the survivorship functions of the two populations do not cross. The equivalence is derived mathematically by linking the optimal transport cost directly to the area between non-crossing survival curves and is checked against data on male and female mortality from 1990 to 2020. A sympathetic reader would care because the result lets a familiar summary statistic, the e0 gap, serve as a ready-made measure of how different two full lifespan distributions are.

Core claim

When the survivorship functions of two populations do not cross, the Wasserstein distance between their age-at-death distributions is exactly equal to the absolute difference in their life expectancies at birth. This identity follows because the non-crossing condition makes the cumulative difference in survival probabilities coincide with the minimal cost of transporting probability mass from one death-age distribution to the other.

What carries the argument

The non-crossing condition on survivorship functions, which ensures the optimal transport plan aligns with the integral of survival differences that defines the e0 gap.

If this is right

In comparisons satisfying the non-crossing condition the e0 gap can be read directly as a complete measure of distributional difference rather than merely a mean difference.
The equivalence holds for standard demographic data such as gender-specific mortality schedules in the Human Mortality Database from 1990 to 2020.
Optimal-transport distances acquire a concrete demographic reading as life-expectancy differences under the stated condition.

Where Pith is reading between the lines

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

Researchers could substitute e0 gaps for full Wasserstein calculations in many routine mortality comparisons once the non-crossing condition is verified.
Similar identities might be tested for other summary measures such as the modal age at death or variance of age at death under analogous restrictions.
The link supplies a bridge that lets transport-based tools be applied to classic questions of lifespan inequality without additional computational overhead.

Load-bearing premise

The survivorship functions of the two populations do not cross.

What would settle it

Two age-at-death distributions whose survivorship functions cross at any age yet still produce a Wasserstein distance exactly equal to their e0 gap, or distributions whose survivorship functions never cross yet produce unequal values.

read the original abstract

The Wasserstein distance is a metric for assessing distributional differences. The measure originates in optimal transport theory and can be interpreted as the minimal cost of transforming one distribution into another. In this paper, the Wasserstein distance is applied to life table age-at-death distributions. The main finding is that, under certain conditions, the Wasserstein distance between two age-at-death distributions equals the corresponding gap in life expectancy at birth ($e_0$). More specifically, the paper shows mathematically and empirically that this equivalence holds whenever the survivorship functions do not cross. For example, this applies when comparing mortality between women and men from 1990 to 2020 using data from the Human Mortality Database. In such cases, the gap in $e_0$ reflects not only a difference in mean ages at death but can also be interpreted directly as a measure of distributional difference.

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.

Desk Editor's Note private letter to a colleague

This paper shows that Wasserstein distance on age-at-death distributions equals the e0 gap exactly when survivorship functions do not cross.

read the letter

The key takeaway is that this paper shows the Wasserstein distance on age-at-death distributions equals the gap in life expectancy at birth when survivorship functions do not cross. It derives this identity directly from the definitions. In one dimension, the Wasserstein-1 distance is the integral of the absolute difference between the cumulative distribution functions. For life tables, that translates to the integral of |S1(x) - S2(x)| where S are the survivorship functions. The e0 difference is the integral of S1(x) - S2(x). These are identical precisely when the difference does not change sign. The paper presents the math cleanly and backs it with an empirical check using Human Mortality Database data from 1990 to 2020. The male-female comparisons in that period meet the non-crossing condition, so the equality works out in the examples. This connection is new in the sense that it hasn't been spelled out before in the demographic literature, even if it follows from standard formulas once the condition is imposed. It gives a way to interpret e0 gaps as measures of distributional difference via optimal transport. The main soft spot is the reliance on the non-crossing assumption. In cases where survivorship curves do cross, the Wasserstein distance exceeds the e0 gap. The paper states this limitation upfront, so it's not a flaw in the argument, but it narrows the practical scope. The result is more of an observation about when two common metrics coincide than a broad new method. Overall, the math and data handling look solid with no obvious circularity or fitting problems. This paper would interest demographers and statisticians working on mortality analysis and distributional metrics. A reader focused on life expectancy trends or applications of Wasserstein distance in social sciences would get value from the explicit link. It deserves a serious referee because the claim is well-supported and the topic bridges two fields in a useful way. I recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims that the 1-Wasserstein distance between two age-at-death distributions equals the absolute difference in life expectancy at birth (e0) whenever the survivorship functions do not cross. This identity is derived from the integral forms W1 = ∫ |S1(x) − S2(x)| dx and |e0_1 − e0_2| = |∫ (S1(x) − S2(x)) dx|, which coincide under the non-crossing premise; the result is illustrated empirically with Human Mortality Database data for 1990–2020, including gender comparisons.

Significance. If the result holds, it supplies a direct, parameter-free link between a core demographic summary measure and the Wasserstein metric from optimal transport, permitting the e0 gap to be read as a distributional distance when survivorship curves do not cross. The derivation rests on standard definitions rather than ad-hoc assumptions, and the empirical check on real mortality schedules adds practical value for interpreting mortality differences.

minor comments (2)

[§3] §3: the notation for the survivorship functions S(x) and the age-at-death densities could be introduced with a brief reminder of their relationship to the cumulative distribution function to aid readers less familiar with life-table conventions.
[Empirical illustration] The empirical section would benefit from a short table or figure caption explicitly listing the country-period pairs examined and confirming that none exhibit crossing survivorship curves in the displayed comparisons.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for the accurate summary of our main result, and for the positive recommendation to accept. No major comments were raised that require a point-by-point reply.

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct mathematical identity

full rationale

The central result equates the 1D Wasserstein-1 distance to the absolute gap in e0 under the explicit non-crossing condition on survivorship functions. This identity follows immediately from the standard integral definitions W_1 = ∫ |S1(x) − S2(x)| dx and |e0_1 − e0_2| = |∫ (S1(x) − S2(x)) dx|, which coincide precisely when S1 − S2 does not change sign. The paper states the non-crossing premise in the abstract and §3, derives the equality under it, and applies it to data satisfying the premise. No step defines one quantity in terms of the other, renames a fitted parameter as a prediction, or relies on a self-citation chain for the load-bearing argument. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definitions of the 1-Wasserstein distance and life expectancy at birth plus the external non-crossing assumption on survivorship functions. No free parameters are introduced and no new entities are postulated.

axioms (2)

standard math Wasserstein distance is defined via the optimal transport formulation between probability measures on the real line.
Invoked in the opening paragraphs to set up the metric.
standard math Life expectancy at birth e0 is the integral of the survivorship function.
Standard demographic identity used to equate the two quantities.

pith-pipeline@v0.9.0 · 5668 in / 1319 out tokens · 32759 ms · 2026-05-18T21:56:13.572649+00:00 · methodology

On the relationship between the Wasserstein distance and differences in life expectancy at birth

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)