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This paper claims that structural non-identifiability in ODE models can be handled directly by building MCMC proposals that respect the geometry of observationally equivalent parameter sets, and it provides two samplers—a geometric MCMC and

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Two MCMC algorithms exploit the geometry of non-identifiable manifolds to speed up posterior sampling in ODE models, with demonstrated efficiency gains on SI and HIV models.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection The pseudo-marginal sampler is a solid contribution, but the geometric sampler's proof contradicts its own algorithm, and that invalidates the paper's central claim. the 4 major comments →

arxiv 2607.13243 v1 pith:NMP7RZZO submitted 2026-07-14 stat.ME cs.NAmath.NA

MCMC Methods for Parameter Inference in Structurally Nonidentifiable Models

Xuyuan Wang , Donglin Han , Michael Y. Li This is my paper
classification stat.ME cs.NAmath.NA MSC 62F1562M0560J0565C05
keywords structural non-identifiabilityBayesian parameter inferenceMarkov chain Monte Carlogeometric MCMCpseudo-marginal MCMCidentifiable parameter combinationsRATTLE constrained integratorrational ODE models
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Structural non-identifiability occurs when different parameter values of an ODE model produce identical output, so the posterior collapses onto lower-dimensional 'equivalence' manifolds that standard MCMC mixes across very slowly. The paper's claim is that this is not an obstacle if one first computes the rational map ξ that sends parameters to identifiable parameter combinations: the posterior factorizes into a marginal on those combinations and a conditional on the equivalence manifold. On that factorization it builds two samplers—a geometric MCMC that teleports along the manifold and jumps across manifolds, and a pseudo-marginal MCMC that samples the identifiable combinations and reconstructs full parameter vectors by resampling. Both are claimed to target the true posterior, to be ergodic, and in case studies on SI and HIV models to improve effective sample sizes by one to two orders of magnitude over random-walk MCMC.

Core claim

The central discovery is a posterior factorization for structurally non-identifiable rational ODE models: writing the identifiable-combination map as ξ(θ)=c, the posterior density π(θ) splits as π_C(c) π(θ|c), where the marginal lives on the low-dimensional space of identifiable combinations and the conditional lives on the observationally equivalent level set M_c={θ:ξ(θ)=c}. The paper uses this split twice. In the geometric MCMC, a RATTLE-based teleportation map proposes moves along M_c and an anisotropic Gaussian proposes moves transverse to it, and a Metropolis correction with an explicit acceptance ratio is claimed to target π; Theorem 3.1 claims uniform geometric ergodicity under compac

What carries the argument

The central object is the identifiable-combination map ξ:Θ→C, obtained from structural identifiability analysis. Its level sets M_c={θ:ξ(θ)=c} are the observationally equivalent submanifolds on which the likelihood is constant. The coarea formula decomposes the posterior into a marginal π_C(c) and a conditional π(θ|c), and both algorithms exploit that decomposition. The geometric sampler relies on a teleportation map T_M built from repeated RATTLE steps—a constrained symplectic integrator that keeps states on M_c—composed with an anisotropic Gaussian proposal that moves in normal and tangent directions. The pseudo-marginal sampler relies on an unbiased importance-sampling estimator of π_C, w

Load-bearing premise

The whole acceptance step for the geometric sampler rests on the assumption that the teleportation map built from repeated constrained RATTLE steps is reversible and preserves volume exactly on the level-set manifold, so the map's Jacobian is 1; the proof as written establishes volume preservation only for the full phase-space map, not for the induced position map on the manifold, so this assumption could fail.

What would settle it

Numerically compute the Jacobian determinant of the M-step teleportation map T_M restricted to the one-dimensional level set of the SI model with ξ fixed. If the determinant is not exactly 1 for some M, the acceptance formula misses a Jacobian factor and the geometric sampler no longer targets the correct posterior. A simpler behavioural test: compare the stationary distribution of Algorithm 2 against Algorithm 4 on the same SI model; if the geometric chain's posterior differs from the pseudo-marginal's, or its acceptance rate drops sharply as M increases, the volume-preservation premise fails

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any rational ODE model whose identifiability analysis yields a rational ξ, posterior inference can be organized around identifiable combinations rather than the full parameter vector, removing the degeneracy that slows standard samplers.
  • The pseudo-marginal sampler's total error splits into two independent components—mixing error in the identifiable-combination coordinates and O(N^{-1/2}) fibre-reconstruction error—so chain iterations and auxiliary sample size can be tuned separately.
  • Because the likelihood is constant along each level set, the pseudo-marginal estimator avoids repeated forward-model evaluations inside the sampler, a meaningful saving when the ODE solver is expensive.
  • On the SI and HIV benchmarks, non-identifiable parameters receive diffuse posteriors as theory requires, while identifiable combinations are recovered tightly; the samplers encode the correct qualitative distinction between identifiable and non-identifiable directions.

Where Pith is reading between the lines

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

  • My inference: the geometric teleportation move and the pseudo-marginal fibre sampler could be combined—using RATTLE teleportation inside each fibre to generate conditional samples when global polynomial root solving becomes too expensive, at the cost of losing the disconnected-components coverage that the global solver provides.
  • My inference: the factorization suggests a reporting standard: for structurally non-identifiable parameters, present posterior summaries over identifiable combinations and conditional fibres, and label point estimates of individual non-identifiable parameters as prior-driven rather than data-driven.
  • My inference: a natural stress test is to run both algorithms on a model with a disconnected level set; the pseudo-marginal sampler's global root solver should explore all components, while the geometric sampler may require an additional component-switching proposal.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper develops two MCMC methods for Bayesian inference in ODE models with structural non-identifiability. The first method, Identifiability-Aware Geometric MCMC, uses a RATTLE-based teleportation map to move along non-identifiable manifolds and an anisotropic Gaussian transition kernel to move between manifolds; Algorithm 2 implements it. The second method, Identifiability-Aware Pseudo-Marginal MCMC, samples in the space of identifiable parameter combinations through an unbiased estimator of the marginal posterior and reconstructs full parameter samples by importance resampling; Algorithm 4 implements it. The paper claims both methods target the correct posterior and are ergodic (Theorems 3.1 and 3.2) and reports numerical gains on SI and HIV models.

Significance. The problem addressed is important: structural non-identifiability is common in mechanistic ODE models and severely degrades MCMC mixing. The pseudo-marginal construction is a genuinely useful formalization: Lemma 3.2 gives an explicit unbiased estimator, Lemma 3.5 gives an O(N^{-1/2}) reconstruction bound, and Theorem 3.2 provides a clean decomposition of marginal-chain error and reconstruction error. The numerical experiments are well-structured and show substantial ESS improvements over random-walk MCMC. The geometric MCMC idea is appealing and, if correctly implemented and analyzed, could be a practical contribution. However, the theoretical analysis of the geometric sampler is invalid as written, so the paper's central correctness claim for one of its two methods is not established.

major comments (4)
  1. [§3.1, Lemma 3.1, Eq. (11)–(12), Algorithm 2] The proposal order in Lemma 3.1 is opposite to the order implemented in Algorithm 2. The lemma defines the proposal as first drawing θ̃∼P(θ,·) and then setting θ′=T_M(θ̃); Algorithm 2 (lines 3 and 7) first sets θ̃=T_M(θ_k) on M_{c_k} and then draws θ′∼qP(·|θ̃). Eq. (12) is therefore the acceptance probability for a different kernel from the one being run. Moreover, T_M is defined only on M_c, while qP has full support in Θ; in the lemma's order, T_M(θ̃) is undefined for θ̃∉M_c. Thus Lemma 3.1 does not establish correctness of the implemented sampler.
  2. [§3.1, Lemma 3.1 proof] The volume-preservation argument proves that the RATTLE phase-space map on D={(θ,p): ξ(θ)=c, Dξ(θ)p=0} preserves the canonical volume. The acceptance formula, however, requires T_M to preserve the Hausdorff measure on the configuration manifold M_c, i.e., |det DT_M|=1 as a map from M_c to itself. This is not shown and is generically false for a projection-based position update such as Algorithm 1's Step 3. The added statement that 'compositions of reversible maps remain reversible' is also not generally true; it requires commutativity, which is not established.
  3. [Algorithm 1, Theorem 3.1] T_M is a stochastic map: Algorithm 1 samples a fresh tangent velocity at each RATTLE step. Therefore T_M is a Markov kernel, not a deterministic map, and the change-of-variables formula Q(θ,dθ′)=qP(T_M^{-1}(θ′)|θ)dθ′ with determinant |det DT_M|=1 is not a valid derivation. In the order used in Lemma 3.1, Q is supported on M_c, a Lebesgue-null subset of Θ, so the subsequent bound Q(θ,dθ′)≥q_min dθ′ with Lebesgue measure is untenable. Consequently the Doeblin minorization and the uniform geometric ergodicity asserted in Theorem 3.1 do not follow.
  4. [§3.2, before Theorem 3.2] The unnormalized marginal is redefined before Theorem 3.2 as ~π_C(c):=∫_{M_c}~π(θ)dθ, omitting the 1/Jξ factor that appears in Eq. (6), Eq. (14), and Lemma 3.2. If dθ is the Hausdorff measure, this integral defines a different marginal from the one the estimator targets; if dθ means something else, the statement is ambiguous. Since Theorem 3.2's invariant measure and error bounds depend on which marginal is being sampled, this needs to be corrected or clarified.
minor comments (4)
  1. [Throughout] Algorithm labels are duplicated: Algorithm 2 is used both for the geometric MCMC sampler and for the coordinate-partition estimator in Section 3.2. The algorithms should be renumbered.
  2. [§4.2, Figure 5c] The identifiable combination for the HIV model is λN/c, but Figure 5c and its caption state 'λc/N'. This typo should be fixed.
  3. [Eq. (19)] The Gaussian notation has an extra comma: 'N(ρI(t_i;θ), , σ²)' should be 'N(ρI(t_i;θ), σ²)'.
  4. [§4.1, Table 1] The paper reports ESS, IACT, and MCSE/SD from what appears to be a single run per method. Since these quantities have substantial Monte Carlo variability, reporting multiple independent chains or uncertainty intervals for the diagnostics would strengthen the numerical claims.

Circularity Check

0 steps flagged

No significant circularity: core derivations are self-contained; the two self-citations are not load-bearing, and the identified proof gaps are correctness issues, not circular reductions.

full rationale

Walking the derivation chain: the identifiability map xi is taken as input from structural identifiability analysis, not derived from the algorithms' outputs. The geometric MCMC acceptance formula (Lemma 3.1) is derived from a change-of-variables argument plus external volume-preservation/reversibility results for RATTLE (Hartmann & Schuette 2005; reviewed in Lelievre et al. 2019). The pseudo-marginal estimator (Lemma 3.2) is unbiased by a direct importance-sampling calculation, and Harris ergodicity is inherited from the standard pseudo-marginal construction of Andrieu and Roberts [2009]. Theorem 3.2(iii)'s bound r(k)+C_k/N^{1/2} is the standard mixing-plus-Monte-Carlo split; no fitted parameter is renamed as a prediction. The numerical sections are synthetic-data benchmarks: data are generated from known ground-truth values, and ESS/IACT diagnostics compare samplers under identical settings. No fitted constant is subsequently called a 'prediction.' The only self-references (Wang 2026; Wang and Han 2026) appear in literature surveys and are not used to justify the central claims, so they are not load-bearing. The Discussion also explicitly states the assumption that identifiability information is available a priori, which is an input assumption, not a circular step. The strongest challenge to the paper is the internal validity of Lemma 3.1/Theorem 3.1: volume preservation is argued on the phase-space set D, while the acceptance ratio needs volume preservation on the configuration manifold M_c; Algorithm 2's proposal order differs from the lemma's order; and Q is supported on a measure-zero set. These are substantive correctness risks, but they are not circular reductions: the proof does not assume its own conclusion, it attempts to derive it from external facts. Therefore they do not raise the circularity score, though they should be weighed as correctness risk.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The methods rely on known structural identifiability analysis and standard MCMC theory. No new scientific entities are introduced. Algorithm hyperparameters are tuning choices, not scientific free parameters.

free parameters (4)
  • RATTLE step size ε = 0.005
    Algorithm hyperparameter in SI example; not derived from theory.
  • Number of RATTLE steps M = 20
    Algorithm hyperparameter; chosen for numerical experiments.
  • Transition proposal scales σ_N, σ_T = 0.8, 0.2
    Chosen to favor moves normal to the manifold; not fitted.
  • Pseudo-marginal auxiliary sample size N = 128
    Number of samples for unbiased estimator; not fitted.
axioms (6)
  • domain assumption The ODE model is rational and admits a rational structural identifiability map ξ = P/Q with generic rank q.
    Used throughout; see Eq. (4).
  • standard math The set where rank(Dξ) < q has Lebesgue measure zero, and level sets M_c are embedded submanifolds almost everywhere.
    Invoked in Section 3, following Hong et al. (2020).
  • standard math Coarea formula and implicit function theorem are applicable.
    Used in Eqs. (6)–(7) and Lemma 3.3.
  • domain assumption Θ is compact and π is continuous and strictly positive for the uniform ergodicity result.
    Assumption in Theorem 3.1.
  • domain assumption For the pseudo-marginal method, the estimator is unbiased with finite variance and the proposal on C is irreducible and aperiodic.
    Assumption 3.1.
  • domain assumption The RATTLE integrator is reversible and volume-preserving on the manifold M_c.
    Used in Lemma 3.1; only established on phase space, not on the manifold.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of MCMC Methods for Parameter Inference in Structurally Nonidentifiable Models." pith.science (2026). https://pith.science/paper/NMP7RZZO

@misc{pith2026260713243,
  author       = {Pith},
  title        = {Pith review of: MCMC Methods for Parameter Inference in Structurally Nonidentifiable Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMP7RZZO}},
  note         = {Machine review of arXiv:2607.13243}
}
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read the original abstract

We consider the problem of parameter inference for ordinary differential equation (ODE) models with structural non-identifiability. Such models arise in a wide range of scientific fields, including control theory, systems biology, and public health. Structural non-identifiability occurs when distinct parameter values provide identical model outputs, resulting in lower-dimensional manifolds of observationally equivalent solutions in the parameter space. This poses challenges for Bayesian inference and Markov chain Monte Carlo (MCMC) methods, often leading to poor mixing and slow convergence. We develop two MCMC methods that use information from structural identifiability analysis. The first, Identifiability-Aware Geometric MCMC, constructs proposals that move within and between non-identifiable manifolds. The second, Identifiability-Aware Pseudo-Marginal MCMC, performs inference on the space of identifiable parameter combinations and reconstructs full parameter values. We show that both methods target the correct posterior distribution and are ergodic under standard conditions. Numerical examples demonstrate improved sampling efficiency and convergence compared with standard MCMC methods.

Figures

Figures reproduced from arXiv: 2607.13243 by Donglin Han, Michael Y. Li, Xuyuan Wang.

Figure 1
Figure 1. Figure 1: Trace plots of the standard random-walk MCMC algorithm fo [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the sampling mechanisms of different MCMC alg [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Trace plots of the proposed identifiability-aware geometr [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of autocorrelation functions (ACFs) of differ [PITH_FULL_IMAGE:figures/full_fig_p041_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of inference results obtained by different MCM [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗

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Reference graph

Works this paper leans on

40 extracted references · 1 linked inside Pith

  1. [1]

    2021 , publisher=

    Feedback systems: an introduction for scientists and engineers , author=. 2021 , publisher=

    2021

  2. [2]

    2016 , publisher=

    Systems biology: a textbook , author=. 2016 , publisher=

    2016

  3. [3]

    1991 , publisher=

    Infectious diseases of humans: dynamics and control , author=. 1991 , publisher=

    1991

  4. [4]

    1995 , publisher=

    Bayesian data analysis , author=. 1995 , publisher=

    1995

  5. [5]

    2004 , publisher=

    Monte Carlo statistical methods , author=. 2004 , publisher=

    2004

  6. [6]

    2011 , publisher=

    Handbook of markov chain monte carlo , author=. 2011 , publisher=

    2011

  7. [7]

    Mathematical biosciences , volume=

    System identifiability based on the power series expansion of the solution , author=. Mathematical biosciences , volume=. 1978 , publisher=

    1978

  8. [8]

    Mathematics and Computers in Simulation , volume=

    Global approaches to identifiability testing for linear and nonlinear state space models , author=. Mathematics and Computers in Simulation , volume=. 1982 , publisher=

    1982

  9. [9]

    Bioinformatics , volume=

    GenSSI 2.0: multi-experiment structural identifiability analysis of SBML models , author=. Bioinformatics , volume=. 2018 , publisher=

    2018

  10. [10]

    automatica , volume=

    On global identifiability for arbitrary model parametrizations , author=. automatica , volume=. 1994 , publisher=

    1994

  11. [11]

    Complexity , volume=

    Observability and structural identifiability of nonlinear biological systems , author=. Complexity , volume=. 2019 , publisher=

    2019

  12. [12]

    Bioinformatics , volume=

    Comparison of approaches for parameter identifiability analysis of biological systems , author=. Bioinformatics , volume=. 2014 , publisher=

    2014

  13. [13]

    Systematic biology , volume=

    Identifiability of parameters in MCMC Bayesian inference of phylogeny , author=. Systematic biology , volume=. 2002 , publisher=

    2002

  14. [14]

    2003 , publisher=

    Measurement error and misclassification in statistics and epidemiology: impacts and Bayesian adjustments , author=. 2003 , publisher=

    2003

  15. [15]

    Journal of the American Statistical Association , volume=

    What are the limits of posterior distributions arising from nonidentified models, and why should we care? , author=. Journal of the American Statistical Association , volume=. 2009 , publisher=

    2009

  16. [16]

    Journal of computational physics , volume=

    Manifold learning for parameter reduction , author=. Journal of computational physics , volume=. 2019 , publisher=

    2019

  17. [17]

    Statistics in Medicine , volume=

    Incorporating Additional Evidence as Prior Information to Resolve Non-Identifiability in Bayesian Disease Model Calibration: A Tutorial , author=. Statistics in Medicine , volume=. 2025 , publisher=

    2025

  18. [18]

    Inverse Problems , volume=

    Likelihood-informed dimension reduction for nonlinear inverse problems , author=. Inverse Problems , volume=. 2014 , publisher=

    2014

  19. [19]

    SIAM Journal on Scientific Computing , volume=

    Accelerating Markov chain Monte Carlo with active subspaces , author=. SIAM Journal on Scientific Computing , volume=. 2016 , publisher=

    2016

  20. [20]

    Bulletin of Mathematical Biology , volume=

    Bayesian Identifiability Analysis for Infectious Disease Models: Parameter Reduction and Model Selection , author=. Bulletin of Mathematical Biology , volume=. 2026 , publisher=

    2026

  21. [21]

    Communications on Pure and Applied Mathematics , volume=

    Global identifiability of differential models , author=. Communications on Pure and Applied Mathematics , volume=. 2020 , publisher=

    2020

  22. [22]

    arXiv preprint arXiv:2511.12847 , year=

    Identification-aware Markov chain Monte Carlo , author=. arXiv preprint arXiv:2511.12847 , year=

    arXiv

  23. [23]

    2014 , publisher=

    Identifiability of parametric models , author=. 2014 , publisher=

    2014

  24. [24]

    PloS one , volume=

    Structural identifiability of systems biology models: a critical comparison of methods , author=. PloS one , volume=. 2011 , publisher=

    2011

  25. [25]

    Computer methods and programs in biomedicine , volume=

    DAISY: A new software tool to test global identifiability of biological and physiological systems , author=. Computer methods and programs in biomedicine , volume=. 2007 , publisher=

    2007

  26. [26]

    Bioinformatics , volume=

    GenSSI: a software toolbox for structural identifiability analysis of biological models , author=. Bioinformatics , volume=. 2011 , publisher=

    2011

  27. [27]

    The pseudo-marginal approach for efficient Monte Carlo computations , author=

  28. [28]

    Salmaniw, AP Browning , author=

    Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators: Y. Salmaniw, AP Browning , author=. Journal of Mathematical Biology , volume=. 2025 , publisher=

    2025

  29. [29]

    ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f

    A constrained hybrid Monte-Carlo algorithm and the problem of calculating the free energy in several variables , author=. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f. 2005 , publisher=

    2005

  30. [30]

    Artificial intelligence and statistics , pages=

    A family of MCMC methods on implicitly defined manifolds , author=. Artificial intelligence and statistics , pages=. 2012 , organization=

    2012

  31. [31]

    Numerische Mathematik , volume=

    Hybrid Monte Carlo methods for sampling probability measures on submanifolds , author=. Numerische Mathematik , volume=. 2019 , publisher=

    2019

  32. [32]

    arXiv preprint arXiv:2604.22055 , year=

    A Replica Exchange Markov Chain Monte Carlo Method for Disconnected Implicit Manifolds via Tubular Relaxation , author=. arXiv preprint arXiv:2604.22055 , year=

    Pith/arXiv arXiv

  33. [33]

    2005 , publisher=

    The Numerical solution of systems of polynomials arising in engineering and science , author=. 2005 , publisher=

    2005

  34. [34]

    SIAM Journal on Mathematics of Data Science , volume=

    Random points on an algebraic manifold , author=. SIAM Journal on Mathematics of Data Science , volume=. 2020 , publisher=

    2020

  35. [35]

    SIAM Journal on Scientific Computing , volume=

    Efficient algorithms for computing a strong rank-revealing QR factorization , author=. SIAM Journal on Scientific Computing , volume=. 1996 , publisher=

    1996

  36. [36]

    SIAM Journal on Scientific Computing , volume=

    Active subspace methods in theory and practice: applications to kriging surfaces , author=. SIAM Journal on Scientific Computing , volume=. 2014 , publisher=

    2014

  37. [37]

    velocity

    Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations , author=. Journal of computational Physics , volume=. 1983 , publisher=

    1983

  38. [38]

    Identifiability and Observability in Epidemiological Models-a survey , author=

  39. [39]

    SIAM review , volume=

    On identifiability of nonlinear ODE models and applications in viral dynamics , author=. SIAM review , volume=. 2011 , publisher=

    2011

  40. [40]

    Mathematical biosciences , volume=

    Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input--output equations , author=. Mathematical biosciences , volume=. 2011 , publisher=

    2011

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.