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Genealogical processes of sequential Monte Carlo methods and other non-neutral population models under rapid mutation

Jere Koskela, Paul A. Jenkins, Adam M. Johansen, Dario Spano

TL;DR

The paper addresses how genealogies in non-neutral population models and SMC algorithms converge to the Kingman coalescent under a specific time rescaling, clarifying when and how different resampling schemes affect coalescence. It introduces a restated convergence theorem with explicit holding-time and merger-scale quantities $c_N(\xi,\ell,j;k)$ and $\tau_N(\xi,\ell,j;t)$, and proves convergence under four strong but verifiable mixing assumptions. The authors provide a substantially simplified proof framework via holding-time analysis and non-Markovian-to-Markovian stitching, and correct prior errors in the conditional transition descriptions. The results have practical implications for designing SMC algorithms, showing how edge-case resampling choices influence genealogical degeneracy and the effective speed of coalescence across multinomial, stratified, and stochastic rounding schemes, with a careful note on the limits of comparing schemes through simple moment orders.

Abstract

We show that genealogical trees arising from a broad class of non-neutral models of population evolution converge to the Kingman coalescent under a suitable rescaling of time. As well as non-neutral biological evolution, our results apply to genetic algorithms encompassing the prominent class of sequential Monte Carlo (SMC) methods. The time rescaling we need differs slightly from that used in classical results for convergence to the Kingman coalescent, which has implications for the performance of different resampling schemes in SMC algorithms. In addition, our work substantially simplifies earlier proofs of convergence to the Kingman coalescent, and corrects an error common to several earlier results.

Genealogical processes of sequential Monte Carlo methods and other non-neutral population models under rapid mutation

TL;DR

The paper addresses how genealogies in non-neutral population models and SMC algorithms converge to the Kingman coalescent under a specific time rescaling, clarifying when and how different resampling schemes affect coalescence. It introduces a restated convergence theorem with explicit holding-time and merger-scale quantities and , and proves convergence under four strong but verifiable mixing assumptions. The authors provide a substantially simplified proof framework via holding-time analysis and non-Markovian-to-Markovian stitching, and correct prior errors in the conditional transition descriptions. The results have practical implications for designing SMC algorithms, showing how edge-case resampling choices influence genealogical degeneracy and the effective speed of coalescence across multinomial, stratified, and stochastic rounding schemes, with a careful note on the limits of comparing schemes through simple moment orders.

Abstract

We show that genealogical trees arising from a broad class of non-neutral models of population evolution converge to the Kingman coalescent under a suitable rescaling of time. As well as non-neutral biological evolution, our results apply to genetic algorithms encompassing the prominent class of sequential Monte Carlo (SMC) methods. The time rescaling we need differs slightly from that used in classical results for convergence to the Kingman coalescent, which has implications for the performance of different resampling schemes in SMC algorithms. In addition, our work substantially simplifies earlier proofs of convergence to the Kingman coalescent, and corrects an error common to several earlier results.
Paper Structure (11 sections, 6 theorems, 100 equations, 2 figures)

This paper contains 11 sections, 6 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A restated convergence theorem
  3. Results for particular resampling schemes
  4. Multinomial resampling
  5. Stratified resampling
  6. Stochastic rounding
  7. Ordering of merger probabilities
  8. Results for Particular Resampling Schemes
  9. Multinomial resampling
  10. Stratified resampling
  11. Technical lemmas

Key Result

Theorem 1

Suppose that for any $n \in \mathbb{N}$, $j < k \in \mathbb{N}_0$, $t \in ( 0, \infty )$, any partition $\xi$ of $[n]$ and any $\eta$ such that $\xi \prec \eta$, and any $\ell \in [ N ]_d^{ | \xi | }$, hold $\mathbb{P}$-almost surely, and that for any sufficiently large $N \in \mathbb{N}$ and any $t > 0$. Then, for any fixed $n \in \mathbb{N}$, weakly in the Skorokhod $J_1$ topology on the spac

Figures (2)

  • Figure 1: An example realisation of the interacting particle model along with the corresponding realisation of the genealogical process. Each row is a generation consisting of $N=5$ particles with labels $1, \ldots, 5$. The arrows point in the direction of the time-evolution of the particle system, while the time-index of the genealogical process counts generations in reverse. Edges highlighted in bold form the ancestral tree of the population in generation 0.
  • Figure 2: The ordered weights of four parents lined up against the four stratification intervals used to implement stratified resampling.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • proof
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Proposition 3
  • Remark 4
  • proof : Proof of Proposition \ref{['prop:multinomial']}
  • ...and 6 more