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arxiv: 2604.20422 · v1 · submitted 2026-04-22 · 🧮 math.ST · math.PR· stat.TH

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Likelihood-based inference for birth-death processes with composite birth mechanisms

Marko Lalovic , Nicos Georgiou , Istvan Z. Kiss
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:00 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH MSC 62F1260J80
keywords birth-death processescomposite birth ratesDoob h-transformlikelihood estimationconsistencyasymptotic normalitySIS epidemicunmarked observations
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The pith

Doob h-transform enables consistent likelihood inference for unmarked composite birth-death processes

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

This paper develops a likelihood-based method to estimate parameters in birth-death processes where multiple mechanisms additively contribute to the birth intensity, such as an SIS epidemic with pairwise and higher-order transmission. The data consist of a single aggregate trajectory in which each birth event is unmarked, so the underlying mechanism is latent and must be deconvolved from the state path alone. The authors replace the original process, conditioned on long survival, by its Doob h-transformed Q-process, which is time-homogeneous and ergodic, and derive the associated conditional likelihood. Under this surrogate measure they prove that both the full conditional maximum-likelihood estimator and a quasi-maximum-likelihood estimator are consistent and asymptotically normal, with asymptotic covariances given by the inverse Fisher and inverse Godambe information matrices. The same framework also supplies a one-dimensional test for the presence of any specific higher-order birth mechanism.

Core claim

Under the Doob h-transformed Q-process the conditional maximum likelihood estimator and the quasi-maximum likelihood estimator are consistent and asymptotically normal for the parameters of composite birth rates, with the asymptotic covariance matrices given by the inverses of the Fisher information and Godambe information matrices, respectively. The approach is illustrated on an SIS epidemic model with pairwise and higher-order transmission, where births are unmarked, and a one-dimensional test statistic is derived to detect specific higher-order mechanisms.

What carries the argument

The Doob h-transformed Q-process, a time-homogeneous ergodic Markov chain that serves as an asymptotic surrogate for the original birth-death process conditioned on long-term survival.

If this is right

  • The estimators recover the true birth-rate parameters consistently from a single long unmarked trajectory.
  • Asymptotic normality allows construction of confidence intervals using the inverse Fisher or Godambe matrices.
  • A practical one-dimensional test can determine whether a particular higher-order birth mechanism is active.
  • Inference remains feasible even when the state space is one-dimensional and mechanism labels are absent.

Where Pith is reading between the lines

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

  • The same conditioning-plus-transform technique could be applied to other survival-biased processes in ecology or chemistry.
  • Numerical methods for computing the h-function would extend the approach to larger state spaces.
  • Applied to real epidemic data the method could test whether higher-order interactions improve transmission models beyond pairwise contacts.

Load-bearing premise

The Doob h-transformed Q-process provides a valid time-homogeneous ergodic surrogate for the law of the original process conditioned on long survival.

What would settle it

Simulate long trajectories from the original process conditioned on survival, apply the estimators, and check whether they converge to the true parameter values with the predicted asymptotic covariance as the trajectory length grows.

Figures

Figures reproduced from arXiv: 2604.20422 by Istvan Z. Kiss, Marko Lalovic, Nicos Georgiou.

Figure 1
Figure 1. Figure 1: Higher-order infection mechanisms in the simplicial SIS model with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of a sample path of the composite birth-death process on [0 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulated surviving trajectory for the simplicial SIS model with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Naive unconditional maximum likelihood estimates based on 200 trajectories of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Conditional MLE for unmarked births from 200 survival-conditioned trajectories of the simplicial SIS model, with N = 100 and X0 = 10. The plotted coordinates are b1 = Nβ1 and b2 = N2β2. The black cross marks the true value (b1, b2) = (1.01, 3.70), and the red circle the empirical mean of the estimates. As T increases, the cloud contracts around the truth. where g (+) β (k; θ) := ∇β log λe k(θ) = f(k) β⊤f(k… view at source ↗
Figure 6
Figure 6. Figure 6: Conditional MLE for marked births from 200 survival-conditioned trajectories of the simplicial SIS model, with N = 100 and X0 = 10. The plotted coordinates are b1 = Nβ1 and b2 = N2β2. The black cross marks the true value (b1, b2) = (1.01, 3.70), and the red circle the empirical mean of the estimates. The clouds are tighter than in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Sample means of the three estimators based on 200 surviving trajectories for each [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Testing H0 : β2 = 0 versus H1 : β2 > 0 in the simplicial SIS model. The figure is based on 1000 survival-conditioned trajectories of the simplicial SIS model, with N = 100, X0 = 10, and observation horizon T = 1000. The data-generating parameters under H0 are b1 = Nβ1 = 2.875, b2 = N2β2 = 0, and µ = 1. In each replicate, the standardized statistic Z2,T was computed from the unconstrained conditional MLE to… view at source ↗
read the original abstract

We develop a likelihood-based inference for finite-state birth-death processes with composite birth rates, in which multiple distinct mechanisms contribute additively to the total birth intensity. Our main motivating example is an SIS epidemic model with pairwise and higher-order transmission. The process is observed through a single aggregate trajectory, and in the main setting of interest, birth events are unmarked. This creates a deconvolution problem in event space: the state is one-dimensional, but the mechanism underlying each birth is latent. We formulate the inference under a Doob $h$-transformed $Q$-process, which is time-homogeneous and ergodic and which provides a time-homogeneous asymptotic surrogate for the law of the original process conditioned on long survival. We derive the corresponding conditional likelihood and study both the conditional maximum likelihood estimator and a quasi-maximum likelihood estimator which is based on a simplified working score. Under the Doob-transform law, we prove consistency and asymptotic normality for both estimators, with asymptotic covariance determined by the inverse Fisher and inverse Godambe information matrices, respectively. We also showcase a practical one-dimensional test for the presence of a specific higher-order birth mechanism.

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 paper develops likelihood-based inference for finite-state birth-death processes with composite (additive) birth rates observed via a single unmarked aggregate trajectory. It formulates the problem under a Doob h-transformed Q-process that is claimed to be time-homogeneous and ergodic, serving as an asymptotic surrogate for the original process conditioned on long survival. Conditional MLE and quasi-MLE estimators are derived, with proofs of consistency and asymptotic normality (covariances via inverse Fisher and Godambe matrices) under the transformed law; a one-dimensional test for higher-order birth mechanisms is also presented. The motivating example is an SIS epidemic model with pairwise and higher-order transmission.

Significance. If the Doob-transform validity and transfer of asymptotic results hold, the work supplies a rigorous framework for parameter inference and mechanism deconvolution in unmarked composite birth-death processes. This is particularly relevant for epidemic modeling and population dynamics, where single trajectories without event marking are common, and could enable practical tests for higher-order interactions without requiring multi-dimensional state augmentation.

major comments (2)
  1. [Doob h-transform and asymptotic results section] The central consistency and asymptotic normality claims (abstract and the section deriving the estimators) rest on the Doob h-transformed Q-process being a valid ergodic surrogate for the law conditioned on long survival. For composite birth intensities, the manuscript does not explicitly verify that the eigenfunction h solving the transform preserves the one-dimensional state space while correctly embedding the latent mechanism deconvolution; this step is load-bearing for transferring the Fisher/Godambe information matrices to the original conditional likelihood.
  2. [Quasi-MLE derivation] The quasi-maximum likelihood estimator based on the simplified working score (section on QMLE) uses the Godambe matrix for its asymptotic covariance. It is unclear whether the working score remains unbiased under the composite-rate generator when the latent birth mechanisms are unmarked; a concrete check against the full conditional score would be needed to confirm the sandwich form applies without additional bias terms.
minor comments (2)
  1. [Model setup] Notation for the composite birth intensity (sum of mechanisms) should be introduced earlier and used consistently when defining the generator Q.
  2. [Numerical illustration] The practical test for higher-order mechanisms would benefit from a small simulation study showing power under the transformed law, even if not required for the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of the Doob h-transform and the quasi-MLE. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Doob h-transform and asymptotic results section] The central consistency and asymptotic normality claims (abstract and the section deriving the estimators) rest on the Doob h-transformed Q-process being a valid ergodic surrogate for the law conditioned on long survival. For composite birth intensities, the manuscript does not explicitly verify that the eigenfunction h solving the transform preserves the one-dimensional state space while correctly embedding the latent mechanism deconvolution; this step is load-bearing for transferring the Fisher/Godambe information matrices to the original conditional likelihood.

    Authors: We appreciate the referee drawing attention to this foundational verification. The underlying birth-death process is defined on the finite one-dimensional state space {0, 1, ..., N} with the composite birth intensity entering additively into the total birth rate function. The Doob h-transform is obtained from the principal eigenpair of the infinitesimal generator Q of this chain; because the state space is finite and the chain is irreducible under the standing assumptions, the eigenfunction h is strictly positive and the transformed generator remains a birth-death generator on exactly the same state space. The latent mechanism deconvolution is encoded in the likelihood via the decomposition of the total intensity, and the h-transform applies uniformly to the aggregate process, thereby preserving the transfer of the Fisher and Godambe information matrices. To make the argument fully explicit, we will insert a short dedicated paragraph (with a brief proof sketch) immediately after the definition of the transformed generator, confirming preservation of the state space and ergodicity for composite rates. revision: yes

  2. Referee: [Quasi-MLE derivation] The quasi-maximum likelihood estimator based on the simplified working score (section on QMLE) uses the Godambe matrix for its asymptotic covariance. It is unclear whether the working score remains unbiased under the composite-rate generator when the latent birth mechanisms are unmarked; a concrete check against the full conditional score would be needed to confirm the sandwich form applies without additional bias terms.

    Authors: We thank the referee for this observation on the unbiasedness of the working score. The working score is constructed by retaining only the terms that depend on the total (observed) intensity while discarding cross-derivative contributions that would require knowledge of the latent mechanism labels. Under the Doob-transformed measure the resulting process is a time-homogeneous Markov chain whose transitions are driven solely by the aggregate rates; consequently the expectation of the working score equals the derivative of the expected log-transition probability and is therefore identically zero. This martingale property carries over directly from the full conditional score because the simplification does not alter the compensator of the observed counting process. To address the referee's request for an explicit check, we will add a short calculation in the QMLE section that verifies E[working score] = 0 under the composite generator and confirms that the Godambe sandwich remains valid without extra bias terms. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard ergodic theory applied to Doob transform

full rationale

The paper formulates inference under the Doob h-transformed Q-process as a time-homogeneous ergodic surrogate for the conditioned process, then derives the conditional likelihood and proves consistency/asymptotic normality via inverse Fisher and Godambe matrices. This chain uses standard results for ergodic Markov chains and does not reduce any prediction or estimator to a fitted input by construction, nor does it depend on load-bearing self-citations or imported uniqueness theorems. The composite birth mechanism is incorporated directly into the generator and likelihood without renaming or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the Doob h-transform is a standard tool from probability theory applied to the inference setting.

pith-pipeline@v0.9.0 · 5503 in / 1301 out tokens · 28269 ms · 2026-05-09T23:00:25.770237+00:00 · methodology

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