Practical Identifiability in Modeling

Updated 27 August 2025

Practical identifiability is the ability to uniquely estimate model parameters from realistic, finite, and noisy experimental data.
It employs methods such as sensitivity analysis, Fisher Information Matrix, profile likelihood, and Monte Carlo simulations to assess parameter reliability.
Optimal experimental design and computational tools are crucial to improving identifiability and guiding effective data collection across disciplines.

Practical identifiability refers to the ability to uniquely determine parameter values in a mathematical model from real, finite, and typically noisy experimental data. Unlike structural identifiability, a property of the model’s equations under ideal data, practical identifiability explicitly accounts for data limitations, noise, and experimental design. It is fundamental for calibrating models in the biological, physical, and engineering sciences, as model-based inference is only meaningful if the relevant model parameters can be reliably and precisely estimated from available observations.

1. Core Concepts and Distinction from Structural Identifiability

Practical identifiability addresses whether parameters or functions thereof can be recovered from actual (finite and noisy) experimental data, given all constraints and sources of error inherent to data acquisition. Even structurally identifiable models may exhibit poor practical identifiability if, for example, small parameter changes cause negligible variations in predicted outputs when compared to measurement error or if data are collected in regimes weakly informative for certain parameters (Wieland et al., 2021, Gallo et al., 2020, Saucedo et al., 26 Jan 2024, Wang et al., 2 Jan 2025).

Key conceptual points include:

Structural identifiability is determined solely by the model equations under perfect conditions and may be assessed via symbolic, differential-algebraic, or model-theoretic methods. It is necessary but not sufficient for practical identifiability.
Practical identifiability requires that the confidence intervals for estimated parameters be finite and appropriately narrow, given real, imperfect data (Wieland et al., 2021). Parameters with infinite or extremely large confidence intervals, even when structurally identifiable, are practically unidentifiable.
Regime dependence: Practical identifiability depends on experimental design, the observation window, the data type (e.g., incidence vs. prevalence in epidemiology), and specific features of the observation process.

2. Quantification and Methodologies

Assessment of practical identifiability involves three main classes of techniques, often deployed in combination:

a) Local Sensitivity and Fisher Information Analysis

Compute local sensitivities (the Jacobian of model outputs with respect to parameters) across the experimental design points to construct the Fisher Information Matrix (FIM):

$F(\theta^*) = s(\theta^*)^T s(\theta^*)$

Practical identifiability is secured if $F(\theta^*)$ is invertible (Wang et al., 2 Jan 2025). Near-zero eigenvalues indicate near non-identifiability along the associated parameter directions.

Regularization can be applied along the poorly-identified eigendirections, and experimental design can be optimized to increase the smallest eigenvalue. New metrics (e.g., projection-based coordinate indices) quantitatively rank parameter identifiability (Wang et al., 2 Jan 2025).

b) Profile Likelihood and Monte Carlo Simulation

The profile likelihood method systematically varies each parameter in turn, optimizing all other parameters, and tracks the change in fit quality (e.g., log-likelihood or residual sum of squares):

$\text{PL}(p_i) = \min_{p_{j\neq i}} \chi^2_\text{res}(p)$

Flat profiles or unbounded confidence intervals signal practical non-identifiability (Wieland et al., 2021, Liu et al., 12 Jun 2025).

Monte Carlo approaches fit the model repeatedly to simulated or resampled noisy data, yielding a distribution for each parameter. Practical identifiability is established if the average relative error (ARE) or mean squared error remains below a prescribed noise-dependent threshold (Saucedo et al., 26 Jan 2024, Liyanage et al., 21 Mar 2025, Heitzman-Breen et al., 20 Jun 2025).

c) Information-Theoretic and Bayesian Approaches

Practical identifiability can be characterized by information-theoretic measures such as conditional mutual information (CMI) computed in the Bayesian posterior framework:

$I(\Theta_i; Y \mid \Theta_{-i}, d) = \mathbb{E}_{\Theta_{-i}} \left[I(\Theta_i; Y \mid \Theta_{-i} = \theta_{-i}, d)\right]$

High CMI for a parameter implies that the data are expected to provide substantial information for that parameter, even before an experiment is performed (Bhola et al., 2023).

In complex models such as nonlinear mixed effects (NLME) models, nonparametric distribution comparisons (e.g., via Kolmogorov–Smirnov tests and the overlap index) across alternative hierarchical fits provide a statistical basis for population-level practical identifiability, without assuming a fixed parametric form (Cassidy et al., 27 Jul 2025).

3. Practical Implications and Experimental Design

The practical identifiability of model parameters is tightly linked to experimental design and data characteristics:

Multiple experiments and model theory: Model-theoretic methods quantify the minimal number of independent experiments (with independent initial conditions) required to achieve maximal identifiability. The identifiability “defect” is tracked as experiments are replicated, and the defect’s stabilization pinpoints the point of maximal available identifiability (Ovchinnikov et al., 2020). In many ODE models, additional experiments cease to improve identifiability once this defect plateaus.
Optimal experiment design: Modern frameworks treat experiment optimization as an optimal control problem. External stimuli or control inputs are selected to maximize information content—often via maximizing the minimum eigenvalue of the FIM or minimizing prediction uncertainty. Pontryagin’s Maximum Principle and forward-backward sweep algorithms provide computational means to optimize the design (Liu et al., 12 Jun 2025).
Input profile and time-varying parameters: The introduction of time-varying or “forcing” functions as scaling or replacement for specific parameters can resolve unidentifiability that cannot be alleviated by structural reparameterization alone, particularly in rational function ODE models (Conrad et al., 3 Jul 2024). Such augmentation leverages existing data streams (e.g., environmental variables in ecological or epidemiological models) to enhance parameter separability.
Data type and frequency: Studies in epidemiology demonstrate that incidence data generally provides superior practical identifiability compared to cumulative or prevalence data, particularly when sampled at high frequency and during the epidemic’s peak (Saucedo et al., 26 Jan 2024, Liyanage et al., 21 Mar 2025).
Initial conditions: Fixing or tightly constraining non-observed initial conditions can eliminate local or global identifiability ambiguities, particularly important in compartmental epidemic models (Chen et al., 25 Jun 2024). In some models, ambiguities manifest as symmetries (e.g., between parameters and corresponding initial conditions) that cannot be resolved with observed data alone.

4. Algorithmic Approaches and Computational Tools

The complexity of practical identifiability analysis has motivated the development of several algorithmic tools and methodologies:

Polynomial complexity algorithms: Polynomial arithmetic complexity is achieved for multi-experiment identifiability analysis via randomized (Monte Carlo) algorithms that estimate identifiability defects through input–output relations and transcendence degree computations. Implementations in Julia (using Oscar and Nemo libraries) demonstrate computational advantages over traditional approaches, particularly for large and nonlinear models (Ovchinnikov et al., 2020).
Weak-form approaches: Weak-form parameter estimation (WENDy framework) eschews repeated forward ODE solves by transforming the input–output relation into a set of integral equations amenable to regression. This approach provides computational speedups and robustness to noise, enabling estimation in models with unobserved states and facilitating rapid assessment of identifiability via the (e, q)-criterion (Heitzman-Breen et al., 20 Jun 2025).
Unified frameworks: Recent proposals integrate sensitivity analysis, FIM eigenspectrum ranking, coordinate-wise projection indices, and regularization in a computational pipeline. This framework not only diagnoses practical identifiability failures but also suggests regularizers (imposing constraints on non-identifiable eigendirections) and guides data collection by targeting points that maximally improve identifiability (Wang et al., 2 Jan 2025).
Nonparametric methods for hierarchical models: Nonparametric comparisons of estimated population parameter distributions (using K–S tests and overlap indices) address the distinct identifiability issues of NLME models, extending the concept of practical identifiability from individuals to populations (Cassidy et al., 27 Jul 2025).

Key Algorithms and Metrics Table

Method	Quantifies	Usage Context
Profile Likelihood	Confidence Intervals	ODE/PDE models
FIM Eigenvalue Decomposition	Sensitivity, Redundancy	Any parametric model
Column Subset Selection (CSS)	Informative directions	Large models/robustness
Monte Carlo Simulation	ARE, MSE Distributions	Noisy, nonlinear, or hierarchical settings
Weak-form (WENDy)	Rapid estimation/MSE	ODEs with partial observations
Conditional Mutual Information	Bayesian info gain	Pre-experiment diagnostics
Nonparametric Overlap Index	Population-level distinguishability	NLME/population models

5. Case Studies and Applications

Epidemiological modeling: The interplay of practical identifiability and initial condition specification, data type, and noise level is well documented for SEIR and related compartmental models. In practice, only certain parameter combinations may be estimable with acceptable precision; fixing initial conditions of unobserved compartments can resolve theoretical ambiguities but may not always translate into improved precision in noisy data (Chen et al., 25 Jun 2024, Saucedo et al., 26 Jan 2024, Liyanage et al., 21 Mar 2025).
Systems biology and pharmacometrics: Model reduction by reparameterization to retain only identifiable combinations is advised in biochemical and cell signaling networks with “sloppy” parameter directions (Ovchinnikov et al., 2023, Maclaren et al., 7 Feb 2025). In complex respiratory mechanics models for preterm infants, systematic use of screening (Morris), deterministic sensitivity, and SVD-based subset selection isolates a practical, physiologically interpretable parameter core for reliable data fitting (Foster et al., 15 Jan 2025).
Experiment design and control: In systems where external inputs are tractable (e.g., synthetic biology, experimental neurophysiology, in vitro cell proliferation), the optimal selection of control protocols substantially enhances practical identifiability—notably via exploitation of transient or sensitive dynamical regimes (Liu et al., 12 Jun 2025).

6. Challenges, Limitations, and Future Directions

Experimental Constraints: Realistic estimation of high-order output derivatives required for rank-based identifiability tests is often impeded by noisy data and finite sampling (Villaverde, 9 Oct 2024).
Bias and Robustness: Weak-form and information-theoretic approaches, although efficient and robust, may be subject to bias or reduced resolution under severe measurement noise or when key state variables are unmeasured.
Approximate and Local Results: Many practical identifiability analyses are local (based on a nominal fit) or “design-point” dependent; ensuring that computed identifiability properties are invariant under model parametrization and sampling is nontrivial (Maclaren et al., 7 Feb 2025).
Hierarchical Inference: Identifiability in hierarchical mixed effects or Bayesian models may not coincide at the individual and population levels. Nonparametric, sampling-based diagnostics are critical for disentangling these effects (Cassidy et al., 27 Jul 2025).
Scalability and Automation: As models increase in size and complexity, efficient, scalable algorithms—ideally amenable to parallelization or high-performance computing—are essential for practical identifiability analysis.

Future directions point to deeper integration of optimal experiment design, machine learning–based sensitivity ranking, information-theoretic design diagnostics, and robust uncertainty quantification tools. The explicit interplay between model structure, experimental design, and noise/resolution characteristics remains central to advances in practical identifiability theory and practice.

Practical identifiability, by quantifying the reliability of parameter estimation under all real-world experimental constraints, remains the cornerstone for credible model-based inference in science and engineering. Its accurate assessment, guided by theory, numerical algorithms, and principled experiment design, is essential for drawing trustworthy conclusions from mathematical models.