Particle filtering methods for partially observed branching processes

In\'es M. del Puerto; Manuel Serrano-Pastor; Miguel Gonz\'alez

arxiv: 2605.20987 · v1 · pith:COHEHQWGnew · submitted 2026-05-20 · 📊 stat.CO · math.PR

Particle filtering methods for partially observed branching processes

Miguel Gonz\'alez , In\'es M. del Puerto , Manuel Serrano-Pastor This is my paper

Pith reviewed 2026-05-21 01:59 UTC · model grok-4.3

classification 📊 stat.CO math.PR

keywords particle filteringbranching processesBayesian inferenceepidemic modelingsequential Monte Carloparameter estimationpartially observed processes

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The pith

Liu-West particle filter enables Bayesian parameter estimation for partially observed branching processes in epidemic models.

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

The paper reviews frequentist estimators for partially observed branching processes and then develops sequential Monte Carlo tools for Bayesian inference. It applies the Liu-West particle filter specifically to estimate parameters in an epidemic model cast as a partially observed branching process. The approach is demonstrated by revisiting and extending an existing epidemic example, yielding posterior distributions rather than point estimates alone.

Core claim

The Liu-West particle filter provides a computational tool for performing Bayesian estimation of the parameters of interest for an epidemic model fitted by a partially observed branching process.

What carries the argument

The Liu-West particle filter, a sequential Monte Carlo algorithm that maintains a cloud of weighted particles to approximate the posterior distribution of model parameters while handling the incomplete observations arising from the branching process.

If this is right

Bayesian posteriors become available for parameters that previously had only frequentist estimators.
The same filter can be used to revisit and extend the epidemic data example from reference [8].
Parameter uncertainty is quantified through full posterior distributions instead of point estimates.
The method supplies a practical route to inference when observations of the branching process are incomplete.

Where Pith is reading between the lines

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

The same particle-filter workflow could be tested on other partially observed count processes arising in ecology or population genetics.
Direct comparison of the Liu-West filter against bootstrap or auxiliary particle filters on the same epidemic data would quantify any differences in bias or computational cost.
Embedding the filter inside a larger hierarchical model would allow joint estimation of both process parameters and observation-error parameters.

Load-bearing premise

The branching process model correctly describes the epidemic dynamics and the particle filter approximation introduces no substantial bias or degeneracy in the resulting posteriors.

What would settle it

Generate simulated epidemic trajectories from the branching process with known true parameter values, run the Liu-West filter on the partial observations, and verify whether the credible intervals contain the true values at the nominal coverage rate.

Figures

Figures reproduced from arXiv: 2605.20987 by In\'es M. del Puerto, Manuel Serrano-Pastor, Miguel Gonz\'alez.

**Figure 2.** Figure 2: Case: π = 0.4 and λ = 0.6 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Case: π = 0.6 and λ = 0.97 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Case: π = 0.8 and λ = 1.47 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Case: π = 0.4 and λ = 0.6 with x0 ∼ U{50, . . . , 200} [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

read the original abstract

This paper focuses on the estimation of partially observed branching processes. First, the estimators from a frequentist perspective proposed in the literature are reviewed. The main objective of this paper is to present computational tools based on sequential Monte Carlo methods to perform Bayesian inference for these processes. In particular, the Liu-West particle filter is applied to perform Bayesian estimation of the parameters of interest for an epidemic model fitted by a partially observed branching process. As application, the example given in [8] is revisited and extended.

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

The paper applies the Liu-West particle filter to Bayesian parameter estimation for a partially observed branching process epidemic model and extends an earlier example.

read the letter

The main point is that the authors review frequentist estimators for partially observed branching processes and then demonstrate the Liu-West particle filter as a tool for Bayesian inference on the parameters of an epidemic model cast in that framework. They revisit and extend the example from reference [8]. That is the actual contribution here. It is an incremental application of an established sequential Monte Carlo method rather than a new derivation or framework. What the paper does reasonably well is lay out a computational route for handling missing observations in these integer-valued processes, which matters for epidemic data where not every infection or event is recorded. The setup connects the branching process directly to the epidemic context and shows how the filter can target the posterior. The soft spots are around validation. The abstract gives no detail on effective sample size behavior, degeneracy checks, or direct comparisons against an exact sampler or MCMC on the revisited example. The stress-test concern about bias from the discrete support and high branching variance is worth examining in the full text; if those diagnostics are missing or weak, the reported posteriors could be less reliable than they appear. Minor issues like that can be fixed in revision, but they matter for a methods paper. This is for readers working on statistical inference for stochastic processes or epidemic modeling with incomplete data. Someone already familiar with particle filters would see a concrete use case rather than a broad advance. It deserves a serious referee because the problem is well-motivated and the approach is grounded in existing tools, even if the novelty is modest. I would send it to peer review with a request for stronger filter diagnostics and baseline comparisons.

Referee Report

1 major / 2 minor

Summary. The manuscript reviews frequentist estimators for partially observed branching processes from the literature and then develops sequential Monte Carlo tools for Bayesian inference on these processes. The central contribution is the application of the Liu-West particle filter to perform Bayesian parameter estimation for an epidemic model formulated as a partially observed branching process; the example from reference [8] is revisited and extended.

Significance. If the Liu-West filter is shown to deliver reliable posteriors for the discrete branching-process epidemic model, the work would supply a practical Bayesian computational framework that complements existing frequentist methods and could be useful for epidemic data with partial observations.

major comments (1)

[Application section (revisited example from [8])] Application section (revisited example from [8]): the manuscript applies the Liu-West filter but does not report effective sample size trajectories or degeneracy diagnostics. Because branching processes produce discrete, high-variance offspring counts under partial observation, the shrinkage step may still permit rapid particle collapse; without these diagnostics or a comparison against an exact sampler on the same example, the claim that the reported posteriors are accurate Bayesian estimates remains unverified.

minor comments (2)

[Introduction] The review of frequentist estimators in the introduction should explicitly state which estimators are being extended or improved upon by the SMC approach.
[Methods] Notation for the branching process offspring distribution and observation model should be introduced with a clear table or list of symbols before the particle-filter algorithm is presented.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address the major comment point by point below.

read point-by-point responses

Referee: Application section (revisited example from [8]): the manuscript applies the Liu-West filter but does not report effective sample size trajectories or degeneracy diagnostics. Because branching processes produce discrete, high-variance offspring counts under partial observation, the shrinkage step may still permit rapid particle collapse; without these diagnostics or a comparison against an exact sampler on the same example, the claim that the reported posteriors are accurate Bayesian estimates remains unverified.

Authors: We agree that the inclusion of effective sample size trajectories and degeneracy diagnostics would provide valuable verification of the Liu-West filter's performance in this setting. In the revised manuscript we will add these diagnostics for the revisited example from [8], including time-series plots of effective sample size and a discussion of observed degeneracy levels. This will directly address concerns about potential particle collapse arising from the discrete, high-variance offspring distribution under partial observation. A comparison against an exact sampler is not feasible for this partially observed branching process model, as no tractable exact sampler exists for the same inference task; this is one motivation for employing particle-filter approximations. The added diagnostics, together with the established theoretical properties of the Liu-West filter, will support the reliability of the reported posteriors. revision: yes

Circularity Check

0 steps flagged

No circularity: standard application of Liu-West filter to branching process model

full rationale

The paper reviews existing frequentist estimators for partially observed branching processes and applies the Liu-West particle filter (a standard SMC method) for Bayesian parameter estimation on an epidemic model. It revisits an example from reference [8] but presents no derivation chain that reduces to fitted quantities by construction, no self-definitional steps, and no load-bearing self-citations that force the central result. The approach is an independent computational application of established tools to the model class, with the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable; the approach relies on standard sequential Monte Carlo techniques applied to branching process models.

pith-pipeline@v0.9.0 · 5604 in / 1065 out tokens · 25454 ms · 2026-05-21T01:59:30.989252+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the Liu-West particle filter is applied to perform Bayesian estimation of the parameters of interest for an epidemic model fitted by a partially observed branching process

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Alvarez, E., Bielska, I.A., Hopkins, S. et al. Limitations of COVID-19 testing and case data for evidence-informed health policy and practice. Health Res Policy Sys 21, 11 (2023). https://doi.org/10.1186/s12961-023-00963-1

work page doi:10.1186/s12961-023-00963-1 2023
[2]

Strong approximations for epidemic models

Ball, F., Donnelly, P. Strong approximations for epidemic models. Stochastic Pro- cess. Appl. 55, 1–21 (1995)

work page 1995
[3]

Becker, N. G. Analysis of Infectious Disease Data. Chapman and Hall/CRC Press: Boca Raton, FL, USA (1989)

work page 1989
[4]

Particle learning and smoothing

Carvalho, C.M., Johannes, M.S., Lopes, H.F., Polson, N. Particle learning and smoothing. Statist. Sci., 25(1), 88-106 (2010)

work page 2010
[5]

P., Kanaan, M

Farrington, C. P., Kanaan, M. N., Gay, N. J. Branching process models for surveil- lance of infectious diseases controlled by mass vaccination. Biostatistics, 4, 279-295 (2003)

work page 2003
[6]

Combined parameters and state estimation in simulation-based filtering

Liu, J., West, M. Combined parameters and state estimation in simulation-based filtering. InSequential Monte Carlo Methods in Practice(A. Doucet, N. de Freitas and N. Gordon, eds.) Springer, New York (2001)

work page 2001
[7]

Meester, R., de Koning, J., de Jong, M. C. M., Diekmann, O. Modeling and real-time piction of classical swine fever epidemics. Biometrics 58, 178–184 (2002)

work page 2002
[8]

Meester, R., Trapman, P.: Estimation in branching processes with restricted obser- vations. Adv. in Appl. Probab., 38, 1098–1115 (2006)

work page 2006
[9]

Kvitkoviˇ cov´ a, A., Panaretos, V

A. Kvitkoviˇ cov´ a, A., Panaretos, V. M.; Asymptotic inference for partially observed branching processes. Adv. in Appl. Probab., 43, 1166–1190 (2011)

work page 2011
[10]

Panaretos, V. M. Partially observed branching processes for stochastic epidemics. J. Math. Biol. 54, 645–668 (2007)

work page 2007
[11]

Asymptotic inference for non-supercritical partially observed branch- ing processes

Rahimov, I. Asymptotic inference for non-supercritical partially observed branch- ing processes. Statist. Probab. Lett., 126, 26–32 (2017)

work page 2017
[12]

Statistical inference for partially observed branhing processes with immigration

Rahimov, I. Statistical inference for partially observed branhing processes with immigration. J. Appl. Probab., 54, 82-95 (2017)

work page 2017
[13]

Estimation of the mean in partially observed branching processes with general immigration

Rahimov, I. Estimation of the mean in partially observed branching processes with general immigration. Stat. Inference Stoch. Process., 22, 143–155 (2019)

work page 2019
[14]

Abkowitz, J., Minin, V.N

Xu, J., Koelle, S., Guttorp, P., Wu, C., Dunbar, C., L. Abkowitz, J., Minin, V.N. Statistical inference for partially observed branching processes with application to cell lineage tracking of in vivo hematopoiesis. Ann. Appl. Stat., 13, 2091 - 2119 (2019). Particle filtering methods for partially observed branching processes 11 π= 0.2 π= 0.4 π= 0.6 π= 0.8...

work page 2091

[1] [1]

Alvarez, E., Bielska, I.A., Hopkins, S. et al. Limitations of COVID-19 testing and case data for evidence-informed health policy and practice. Health Res Policy Sys 21, 11 (2023). https://doi.org/10.1186/s12961-023-00963-1

work page doi:10.1186/s12961-023-00963-1 2023

[2] [2]

Strong approximations for epidemic models

Ball, F., Donnelly, P. Strong approximations for epidemic models. Stochastic Pro- cess. Appl. 55, 1–21 (1995)

work page 1995

[3] [3]

Becker, N. G. Analysis of Infectious Disease Data. Chapman and Hall/CRC Press: Boca Raton, FL, USA (1989)

work page 1989

[4] [4]

Particle learning and smoothing

Carvalho, C.M., Johannes, M.S., Lopes, H.F., Polson, N. Particle learning and smoothing. Statist. Sci., 25(1), 88-106 (2010)

work page 2010

[5] [5]

P., Kanaan, M

Farrington, C. P., Kanaan, M. N., Gay, N. J. Branching process models for surveil- lance of infectious diseases controlled by mass vaccination. Biostatistics, 4, 279-295 (2003)

work page 2003

[6] [6]

Combined parameters and state estimation in simulation-based filtering

Liu, J., West, M. Combined parameters and state estimation in simulation-based filtering. InSequential Monte Carlo Methods in Practice(A. Doucet, N. de Freitas and N. Gordon, eds.) Springer, New York (2001)

work page 2001

[7] [7]

Meester, R., de Koning, J., de Jong, M. C. M., Diekmann, O. Modeling and real-time piction of classical swine fever epidemics. Biometrics 58, 178–184 (2002)

work page 2002

[8] [8]

Meester, R., Trapman, P.: Estimation in branching processes with restricted obser- vations. Adv. in Appl. Probab., 38, 1098–1115 (2006)

work page 2006

[9] [9]

Kvitkoviˇ cov´ a, A., Panaretos, V

A. Kvitkoviˇ cov´ a, A., Panaretos, V. M.; Asymptotic inference for partially observed branching processes. Adv. in Appl. Probab., 43, 1166–1190 (2011)

work page 2011

[10] [10]

Panaretos, V. M. Partially observed branching processes for stochastic epidemics. J. Math. Biol. 54, 645–668 (2007)

work page 2007

[11] [11]

Asymptotic inference for non-supercritical partially observed branch- ing processes

Rahimov, I. Asymptotic inference for non-supercritical partially observed branch- ing processes. Statist. Probab. Lett., 126, 26–32 (2017)

work page 2017

[12] [12]

Statistical inference for partially observed branhing processes with immigration

Rahimov, I. Statistical inference for partially observed branhing processes with immigration. J. Appl. Probab., 54, 82-95 (2017)

work page 2017

[13] [13]

Estimation of the mean in partially observed branching processes with general immigration

Rahimov, I. Estimation of the mean in partially observed branching processes with general immigration. Stat. Inference Stoch. Process., 22, 143–155 (2019)

work page 2019

[14] [14]

Abkowitz, J., Minin, V.N

Xu, J., Koelle, S., Guttorp, P., Wu, C., Dunbar, C., L. Abkowitz, J., Minin, V.N. Statistical inference for partially observed branching processes with application to cell lineage tracking of in vivo hematopoiesis. Ann. Appl. Stat., 13, 2091 - 2119 (2019). Particle filtering methods for partially observed branching processes 11 π= 0.2 π= 0.4 π= 0.6 π= 0.8...

work page 2091