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arxiv: 2604.20629 · v1 · submitted 2026-04-22 · 🧮 math.PR · q-bio.PE

Recognition: unknown

Rates of forgetting for the sequentially Markov coalescent

Jonathan Terhorst
Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:46 UTC · model grok-4.3

classification 🧮 math.PR q-bio.PE
keywords sequentially Markov coalescentgeometric ergodicitytotal variation distanceSMC'Markov jump processforgetting rategenetic distancepairwise coalescent
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The pith

The pairwise sequentially Markov coalescent forgets its initial condition geometrically fast in the embedded jump chain but only at rate 1/ℓ in the continuous process.

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

This paper studies how fast the sequentially Markov coalescent loses memory of its starting genealogy when modeling correlations along a chromosome in the pairwise setting. For the discrete embedded jump chain it proves geometric ergodicity to stationarity in total variation distance and supplies explicit constants. The continuous-time version converges more slowly, with the distance to equilibrium decaying asymptotically like one over genetic distance ℓ. The same forgetting behavior holds for the related SMC' process after a time-change argument. These rates supply a theoretical basis for common approximations that treat sufficiently distant loci as independent.

Core claim

For the embedded jump chain of the pairwise SMC we prove geometric ergodicity in total variation with explicit constants. For the continuous process, by contrast, the total variation distance from stationarity decays as ≍ 1/ℓ in genetic distance ℓ. We obtain analogous results for the closely related SMC' process using a novel time-change argument.

What carries the argument

The embedded jump chain of the sequentially Markov coalescent together with a time-change argument relating the continuous SMC and SMC' processes.

If this is right

  • Heuristic approximations in the literature that treat distant loci as evolving independently are justified.
  • Explicit constants allow quantitative error bounds when using SMC-based methods for population history estimation.
  • The same geometric and 1/ℓ rates apply to the SMC' process after the time change.
  • Independence assumptions become reliable once genetic distance exceeds a scale set by the explicit constants.

Where Pith is reading between the lines

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

  • The slow 1/ℓ decay implies that correlation between loci persists farther along the chromosome than geometric forgetting alone would suggest.
  • Numerical checks of the total variation distance at large ℓ would directly test the predicted continuous-time rate.
  • Extending the geometric ergodicity proof beyond the pairwise case would be needed before the same justification applies to full-sample SMC models.
  • The explicit constants could be used to set distance thresholds in practical ancestry-inference algorithms.

Load-bearing premise

The analysis is restricted to the pairwise setting with only two lineages.

What would settle it

A direct computation or simulation that measures total variation distance to stationarity for the continuous SMC at successively larger genetic distances ℓ and finds that the distance does not scale asymptotically as 1/ℓ.

read the original abstract

The sequentially Markov coalescent (SMC) is a Markov jump process which models correlations in local genealogies across a chromosome. It has been used as a theoretical tool for studying linkage disequilibrium and identity-by-descent, and it also forms the basis of a class of statistical procedures for estimating population history and inferring ancestry. In this paper, we study the rate at which SMC forgets its initial condition in the pairwise setting. For the embedded jump chain, we prove geometric ergodicity in total variation, with explicit constants. For the continuous process, by contrast, the total variation distance from stationarity decays as $\asymp 1/\ell$ in genetic distance $\ell$. We obtain analogous results for the closely related SMC' process using a novel time-change argument. One application of these results is to justify heuristic approximations used in the literature that treat distant loci as evolving independently.

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

0 major / 3 minor

Summary. The manuscript establishes the rate at which the pairwise sequentially Markov coalescent (SMC) forgets its initial condition. For the embedded jump chain, it proves geometric ergodicity in total variation with explicit constants. For the continuous process indexed by genetic distance ℓ, the total variation distance to stationarity decays asymptotically as ≍ 1/ℓ. Analogous results for the SMC' process are obtained via a time-change argument. These findings justify approximations that treat distant loci as evolving independently.

Significance. If the proofs hold, the work supplies rigorous quantitative mixing rates for SMC models central to population-genetics analyses of linkage disequilibrium and identity-by-descent. The explicit constants for the jump-chain ergodicity and the precise 1/ℓ decay for the continuous process are practically useful; the time-change argument for SMC' is a clean technical contribution. The direct corollary on distant-loci independence strengthens the justification for widely used heuristic approximations in ancestry inference and demographic estimation.

minor comments (3)
  1. [§2.1] §2.1: the precise functional form of the state-dependent jump rates (proportional to current coalescence time) is used repeatedly but is only sketched; writing the rate function explicitly would aid verification of the subsequent ergodicity arguments.
  2. [Theorem 3.2] Theorem 3.2: the constant C in the geometric ergodicity bound is stated to be explicit, yet its dependence on the initial coalescence time t_0 is not displayed; adding the explicit expression (or a short derivation) would increase usability.
  3. [§4.3] §4.3: the time-change argument for SMC' is elegant, but the measurability of the inverse time-change map with respect to the filtration is asserted without a reference or short proof; a one-sentence justification would remove any doubt.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the explicit constants, the 1/ℓ decay rate, and the time-change argument for SMC'. We note the recommendation for minor revision and will address any editorial or minor technical points in the revised version.

Circularity Check

0 steps flagged

No significant circularity; results follow from direct Markov process analysis

full rationale

The paper proves geometric ergodicity in total variation for the embedded jump chain of the pairwise SMC (with explicit constants) and ≍ 1/ℓ decay for the continuous-time process via direct analysis of the state-dependent jump rates and a time-change argument for SMC'. No load-bearing step reduces to a fitted parameter, self-citation chain, ansatz, or renaming of a known result; the derivations are self-contained from the process definitions. This matches the reader's assessment of direct analysis without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard Markov process theory and the definition of the SMC; no free parameters, invented entities, or ad-hoc axioms are apparent from the abstract.

axioms (1)
  • domain assumption The SMC is a Markov jump process with the standard transition structure for pairwise coalescent events.
    Invoked implicitly in the statement of geometric ergodicity for the embedded chain.

pith-pipeline@v0.9.0 · 5438 in / 1090 out tokens · 17211 ms · 2026-05-09T22:46:04.278623+00:00 · methodology

discussion (0)

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Reference graph

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