Parameter Identifiability beyond the Fisher Information Approximation in Data-Driven Modeling

arXiv:2602.20495 · q-bio.QM · Submitted 2026-02-24 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: Today's paper: "Parameter Identifiability beyond the Fisher Information Approximation in Data-Driven Modeling".

Marcus: The gist The paper presents a computational framework that uncovers fundamental scaling laws governing practical identifiability and uncertainty quantification in data-driven biological modeling,

Ines: First, who's behind it and why it matters.

Paper summary: Ines: So we're talking about this paper now. It's called "Parameter Identifiability beyond the Fisher Information Approximation in Data-Driven Modeling." Marcus and I think it tackles a big problem in making sense of complex biological data.

Marcus: Exactly. The core idea here is that when you try to use data-driven models for things like HIV dynamics or Alzheimer's disease, you need to know if the parameters you estimate are actually stable or if your model can even recover them reliably.

Ines: Right, so they're moving beyond just using the Fisher Information Matrix, which is often a rough approximation. They want a way to rigorously check parameter identifiability under data perturbations while keeping track of the full Hessian matrix.

Marcus: That sounds like they're trying to fix that issue where traditional methods might miss directions where the model is poorly constrained, or where the likelihood profile is flat.

Ines: Their main contribution seems to be a new metric, K i, which quantifies parameter identifiability at different orders. They have this zero-order component, K zero which recovers the traditional parameter identifiability we already know.

Marcus: And they also introduce the first-order metric, K one which specifically captures when you see flat likelihood profiles when that zero-order component is zero <ref:2602.20495#pg1>. That's where things get tricky for people trying to fit parameters with weak constraints.

Ines: It seems like this is crucial because traditional profile likelihood methods often fail when the profiles are numerically insensitive, and this framework aims to resolve those parameters.

Marcus: It’s important because it gives us a way to see exactly which parameters are resolvable at different scales of complexity in the model. They use an eigenvalue decomposition with the Schur complement to classify identifiability across these hierarchical scales.

Ines: So, what does this mean practically for our work? We're looking at HIV signaling, and this framework helps us understand where the parameter uncertainty is coming from—is it just noise, or is it because of these non-identifiable subspaces?

Marcus: And they don't stop there. They build a robust uncertainty quantification protocol that explicitly accounts for higher-order practically nonidentifiable parameters. They separate the contribution from zero-order and first-order nonidentifiability into different regions in the parameter space, red and blue regions.

Ines: That hierarchical containment principle where higher-order uncertainty manifolds are nested within their zero-order counterparts is pretty powerful for understanding how errors propagate through these models.

Marcus: When they look at the HIV host-virus dynamics model, they found that some eigen-coordinates show zero-order practical identifiability, while others display first-order practical identifiability. That's a specific scaling law in action.

Ines: And when we look at amyloid- beta accumulation in the brain, they see similar patterns with higher-order practical identifiability appearing in the carrying capacity parameters across different cortical regions. It’s consistent across these very different biological systems.

Marcus: The paper shows that zero-order non-identifiability influences both the initial HIV outbreak and later phases of viral relapse, while first-order non-identifiability mainly modulates those early stages of infection. That's a specific way the uncertainty scales with time or infection progression.

Ines: So, it’s about providing a diagnostic tool to ensure that when we build these large-scale mechanistic digital twins, we know exactly where our data inferences are grounded and where the model is pushing us toward uncertainty.

Marcus: It really provides a concrete way to move past just relying on Fisher information when dealing with high-dimensional data. The scaling laws they uncover suggest a fundamental structure to parameter resolvability that we can use across many fields.

Ines: So, what's the conclusion of this paper? In short, it lays out this novel framework for analyzing higher-order parameter identifiability analysis based on scaling laws governing parameter resolvability.

Marcus: And it grounds all of this in a computational approach that is applicable to many high-dimensional biological models. It’s about ensuring our data-driven inferences are robust by understanding these scaling laws governing how parameters can actually be resolved.

Ines: It seems like the authors are providing a necessary structure for translating complex, large-scale mechanistic digital twins into something that is actionable and biologically meaningful.

Marcus: That's what it boils down to. They give us a way to quantify uncertainty arising from non-identifiable subspaces across different orders, which helps us manage those risks in our models.

Conclusion: Ines: So we’ve been deep into how they use this new metric K i to check if a parameter is really identifiable in these big biological models, and now we need to wrap up what all this actually means.

Marcus: Yeah, the paper title itself, "Parameter Identifiability beyond the Fisher Information Approximation," it’s pretty direct about what they’re trying to fix—moving past just using that standard Fisher Information Matrix.

Ines: Right. It says they're looking at how parameters are stable even when you mess with the data a little bit, and they do this while keeping track of the whole Hessian matrix instead of just discarding it.

Marcus: And I think that’s where it gets interesting for us in genomics data science—we want to know if our estimates hold up under noise or batch effects, and this method gives us a way to measure that stability directly.

Ines: They introduce these scaling laws, so the paper shows you a hierarchy of how resolvable parameters are, which isn't just one single number but a whole structure across the parameter space.

Marcus: It’s not just about whether we can find all the parameters, it's about *how* difficult it is to find them at different levels of complexity in the model itself.

Ines: And when you look at HIV dynamics or amyloid-beta accumulation, they show these patterns—some parameters are identifiable at the first order, others only after a second order correction.

Marcus: That’s where I get excited about the implications for cohort studies; if we can map out where those non-identifiable directions are, we know exactly what kind of data perturbation would cause us to lose track of our estimates.

Ines: So the authors are giving us a way to diagnose which parts of the biological system we need to worry about most when building these large-scale digital twins.

Marcus: And they also show how uncertainty itself scales, separating the uncertainty coming from zero-order versus first-order nonidentifiable parameters. It’s a systematic way to quantify that risk in our predictions.

Ines: So, the main point is that this framework moves us past simple approximations and gives us a more rigorous tool for checking if our data-driven biology makes sense on a larger scale.

Marcus: Exactly. And it sets up the next step for figuring out how to build those robust digital twins we talked about earlier.

Department of Mathematics, Penn State University

q-bio.QM

Submitted: 2026-02-24

Updated: 2026-10-08

Comments: 53 pages, 5figures

Code: https://github.com/WilliamMoriaty/Scaling-Law-Practical-Identifiability

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist The paper presents a computational framework that uncovers fundamental scaling laws governing practical identifiability and uncertainty quantification in data-driven biological modeling,

Key concepts

Practical Identifiability
This concept defines when a parameter is truly identifiable in practice. It means that for any small amount of data perturbation (ε), there exists a constant threshold (C) such that the resulting change in the estimated parameter remains bounded. This goes beyond standard mathematical identifiability by considering how stable estimates are when faced with real-world noise.
Ki Metric
Ki is a novel metric used to measure the order of identifiability for each parameter. The zero-order component (K0) recovers traditional identifiability, while the first-order component (K1) detects parameters where likelihood profiles are flat. This helps identify parameters that traditional methods miss due to numerical insensitivity.
Hierarchical Structure
The paper shows that parameter resolvability is organized in a hierarchy across different orders of approximation (O(1), O(ε), O(ε²)). This structure dictates which parameters are identifiable at different levels of precision, allowing researchers to understand the complexity and stability inherent in large-scale models.
Uncertainty Quantification (UQ) Protocol
This protocol quantifies predictive uncertainty by isolating contributions from non-identifiable subspaces. It separates uncertainty arising from zero-order and first-order non-identifiable parameters, showing how different parameter groups influence the model's prediction error at various stages of the simulation.

Terminology

Summary

The gist The paper presents a computational framework that uncovers fundamental scaling laws governing practical identifiability and uncertainty quantification in data-driven biological modeling, which is critical for ensuring that data-driven inferences are grounded in ver1.

How it works

The framework begins by considering a general model φ(t, θ) where t denotes the independent input variable and θ ∈ Rk is the parameter vector <ref:2602.20495#pg6>. Using the least-squares objective l(h(t, θ), hˆ) = PN i=1∥h(ti, θ) − hˆ i∥22 obtains the optimal parameter set θ∗ and computes the Fisher Information Matrix (FIM, F) and the perturbed Hessian (H) <ref:2602.20495#pg7>. The core of the method involves integrating eigenvalue decomposition (EVD) with the Schur complement (SC) to classify parameter identifiability across hierarchical scales <ref:2602.20495#pg7>.

The paper introduces a novel metric, Ki, to quantify the i-th order of parameter identifiability <ref:2602.20495#pg6>. The zero-order component, K0, recovers traditional parameter identifiability as K0 =∥(I − AA†)si∥22 <ref:2602.20495#pg6>, while the first-order metric, K1, captures the emergence of flat likelihood profiles when K0 = 0 <ref:2602.20495#pg6>. This approach is designed to resolve parameters characterized by relatively flat likelihood profiles that traditional profile likelihood methods often fail to resolve due to numerical insensitivity <ref:2602.20495#pg5>.

Coordinate Identifiability and Uncertainty Quantification

The framework rigorously quantifies practical identifiability through the definition of practical identifiability (Definition 1) which states that a parameter θ in Θ is practically identifiable if ∀ε > 0, ∃C > 0 such that∥δ∥. The asymptotic analysis utilizes the relationship between the FIM and practical identifiability (27) and extends this using Schur Complement Theory to provide an asymptotic analysis framework of practical identifiability <ref:2602.20495#pg18>.

The framework establishes a hierarchical structure for parameter resolvability through Theorem 1, which expresses the matrix F + εH approximately as a third-order expansion in ε <ref:2602.20495#pg19>. This theorem displays the practically identifiable and non-identifiable parameters at multiple ε-order, detailing zero-order epsilon (O(1)), first-order epsilon (O(ε)), and second-order epsilon (O(ε2)) classifications <ref:2602.20495#pg19>.

Scaling Laws and Applications

The analysis reveals a complex spectral hierarchy in the parameter space, as shown in Figure 2C <ref:2602.20495#pg8>. For the canonical HIV infection model, the results reveal a hierarchical structure where eigen-coordinates U⊤1θ, U⊤2θ, and U⊤3θ exhibit zero-order practical identifiability, whereas U⊤4θ and U⊤5θ display first-order practical identifiability <ref:2602.20495#pg10>.

The framework was validated on two distinct high-dimensional paradigms: HIV host-virus dynamics and the spatiotemporal propagation of Amyloid-β (Aβ) in the brain <ref:2602.20495#pg14>. In the HIV model, higher-order parameter identifiability is prominently observed in the virion production rate (π) of infected CD4+ T cells and the viral clearance rate (c) <ref:2602.20495#pg10>. For Aβ dynamics, higher-order practical identifiability is evident in the carrying capacity parameters (K) across individual cortical regions <ref:2602.20495#pg12>.

Uncertainty Quantification

The framework also establishes a robust UQ protocol that explicitly accounts for higher-order practically nonidentifiable parameters <ref:2602.20495#pg6>. The uncertainty quantification evaluates predictive uncertainty arising from non-identifiable subspaces by isolating contributions from zero-order non-identifiable parameters (U⊤k−r0θ, red region) and first-order non-identifiable parameters (U⊤k−r0−r1θ, blue region) <ref:2602.20495#pg7>.

The UQ analysis demonstrates a hierarchical containment principle in which higher-order uncertainty manifolds are nested within their zero-order counterparts <ref:2602.20495#pg14>. Specifically, the uncertainty of zero-order ε (l(0)unc) is computed as 1/2 σ2ε · Tr(Σr1 + B1), and the variance of the loss l(0)unc is 1/2 σ4ε2 · Tr((Σr1 + B1)2) <ref:2602.20495#pg25>.

The uncertainty of first-order ε (l(1)unc) is computed as 1/2 σ2ε2 · Tr(S2), and the variance of the loss l(1)unc is 1/2 σ4ε4 · Tr(S22) <ref:2602.20495#pg25>. This demonstrates that parameters exhibiting zero-order non-identifiability exert a broad influence over both the initial HIV outbreak and subsequent phases of viral relapse, while those characterized by first-order non-identifiability primarily modulate the early stages of infection <ref:2602.20495#pg11>.

The framework provides a critical diagnostic tool to ensure that data-driven insights are translated into robust, actionable, and biologically meaningful knowledge in the era of large-scale mechanistic digital twins <ref:2602.20495#pg2>. The paper summarizes a novel framework for higher-order parameter identifiability analysis grounded in scaling laws that govern parameter resolvability <ref:2602.20495#pg15>.

How it works

The paper introduces a novel metric, Ki, to quantify the i-th order of parameter identifiability <ref:2602.20495#pg6>. The zero-order component, K0, recovers traditional parameter identifiability as K0 =∥(I − AA†)si∥22 <ref:2602.20495#pg6>, while the first-order metric, K1, captures the emergence of flat likelihood profiles when K0 = 0 <ref:2602.20495#pg6>. This approach is designed to resolve parameters characterized by relatively flat likelihood profiles that traditional profile likelihood methods often fail to resolve due to numerical insensitivity <ref:2602.20495#pg5>.

The framework rigorously quantifies practical identifiability through the definition of practical identifiability (Definition 1) which states that a parameter θ in Θ is practically identifiable if ∀ε > 0, ∃C > 0 such that∥δ∥. The asymptotic analysis utilizes the relationship between the FIM and practical identifiability (27) and extends this using Schur Complement Theory to provide an asymptotic analysis framework of practical identifiability <ref:2602.20495#pg18>.

How it works

The framework establishes a hierarchical structure for parameter resolvability through Theorem 1, which expresses the matrix F + εH approximately as a third-order expansion in ε <ref:2602.20495#pg19>. This theorem displays the practically identifiable and non-identifiable parameters at multiple ε-order, detailing zero-order epsilon (O(1)), first-order epsilon (O(ε)), and second-order epsilon (O(ε2)) classifications <ref:2602.20495#pg19>.

The analysis of coordinate practical identifiability is defined using the Bayesian posterior likelihood (34, 16, 36) as follows <ref:2602.20495#pg19>. The minimization of loss function L(θi, θ˜; ε) to θ˜ satisfies the necessary condition: 2b(ε)θi + 2c(ε)θ˜ = 0, θ˜ = −c(ε)†b(ε)θi <ref:2602.20495#pg19>. The resulting loss L(θi; θ˜, ε) is rewritten as L(θi; θ˜, ε) ≈ K0 + K1ε + K2ε2 <ref:2602.20495#pg27>.

How it works

The framework also establishes a robust UQ protocol that explicitly accounts for higher-order practically nonidentifiable parameters <ref:2602.20495#pg6>. The uncertainty quantification evaluates predictive uncertainty arising from non-identifiable subspaces by isolating contributions from zero-order non-identifiable parameters (U⊤k−r0θ, red region) and first-order non-identifiable parameters (U⊤k−r0−r1θ, blue region) <ref:2602.20495#pg7>.

Improvements for AI systems

  1. textbf Higher-order parameter identifiability for HIV dynamics in precision therapy selection.: The framework allows for higher-order practical identifiability, which is prominently observed in the virion production rate (π) of infected CD4+ T cells and the viral clearance rate (c) to delineate specific pathophysiological phases of HIV infection governed by distinct parameter subsets.

  2. textbf Spatiotemporal uncertainty quantification for Alzheimer’s disease progression.: The model can now resolve uncertainty by showing that higher-order practical identifiability is evident in the carrying capacity parameters (K) across individual cortical regions, enabling clinicians to distinguish between the terminal saturation state and the transient phase of rapid Aβ accumulation.

  3. textbf Robust parameter inference under flat likelihood manifolds.: The system can now identify parameters where traditional methods fail, as it reveals that the emergence of flat likelihood profiles is primarily governed by the first-order metric K1, allowing for accurate resolution in cases where conventional profile likelihood methods are often infeasible due to prohibitive computational costs.

  4. textbf Hierarchical uncertainty mapping for actionable clinical predictions.: The UQ framework enables the construction of confidence intervals that hierarchically nest within their zero-order counterparts, providing a rigorous way to assess how parameter ambiguity propagates, which is essential for translating model outputs into actionable precision medicine tools and clinical decision-support systems.

Abstract

Data-driven mechanistic modeling of high-dimensional biological data requires rigorous parameter identifiability. Profile likelihood entails costly repeated optimization, while Fisher-information criteria discard the model's Hessian and may misjudge weakly constrained directions. From a well-posedness definition of identifiability we characterize the stability of the estimated parameters under data perturbations while retaining the full Hessian. It yields Hölder- 12 stability for a zero residual when the Fisher information is degenerate and corrects the Fisher matrix otherwise; residual and perturbation sizes decide the order of stability, Lipschitz below a crossover scale proportional to the residual norm and Hölder- 12 above. In a local quadratic approximation we derive coordinate identifiability metrics allowing singular Jacobians and nuisance Hessian blocks, and spectral bounds link the corrections to parameter and prediction uncertainty. Applications to JAK--STAT signaling, HIV dynamics, and amyloid- β accumulation support the theory. Requiring only local derivatives, the framework offers a tractable basis for identifiability analysis of large-scale models.

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