Identifiability
Identifiability asks whether observations contain enough information to determine the unknown parameters of a mathematical model. A model may reproduce the data extremely well while some of its fitted parameter values remain undetermined.
Model outputs and parameters
Let a model with parameter vector \(\boldsymbol\theta\) produce observable output
\[\mathbf y(t;\boldsymbol\theta).\]If two parameter vectors satisfy
\[\mathbf y(t;\boldsymbol\theta^{(1)})=\mathbf y(t;\boldsymbol\theta^{(2)})\]for all relevant observations, then the data cannot distinguish those parameter vectors.
Global structural identifiability
A parameterisation is globally structurally identifiable when equality of ideal model outputs implies equality of the parameters:
\[\boxed{\mathbf y(t;\boldsymbol\theta^{(1)})=\mathbf y(t;\boldsymbol\theta^{(2)})\ \forall t\quad\Longrightarrow\quad\boldsymbol\theta^{(1)}=\boldsymbol\theta^{(2)}}.\]This is an ideal mathematical property of the model together with the observation scheme.
Local structural identifiability
Sometimes ideal observations determine only a finite number of parameter possibilities rather than one unique global value.
The model may then be locally identifiable but not globally identifiable.
Structural non-identifiability
A model is structurally non-identifiable when different parameter values can produce exactly the same ideal observations.
No increase in measurement precision can resolve this without changing what is observed, fixing parameters, reparameterising the model or adding new information.
A simple example
Suppose the observable model is
\[y(t)=ab e^{-t}.\]Only the product \(ab\) affects the output. For any non-zero constant \(c\),
\[(a,b)\quad\text{and}\quad(ca,b/c)\]produce exactly the same trajectory.
Therefore \(a\) and \(b\) cannot be identified separately from observations of \(y(t)\), although the combination \(ab\) can be identified.
Identifiable parameter combinations
Structural non-identifiability does not always mean that nothing can be learned.
A combination such as
\[\phi=ab\]may be identifiable even when \(a\) and \(b\) individually are not.
Reparameterising the model in terms of identifiable combinations can produce a more meaningful inference problem.
Observation scheme matters
Identifiability is not a property of the differential equations alone. It depends on which states or functions of states are observed.
A model may be identifiable when two biological variables are measured but non-identifiable when only one is observed.
Unknown initial conditions
Initial conditions may also interact with model parameters.
If \(x(0)=x_0\) is unknown, then \(x_0\) should be treated as part of the inference problem. An unknown initial state can introduce additional non-identifiability.
Practical identifiability
Even when a parameter is structurally identifiable, real observations may determine it poorly because data are noisy, sparse, collected over too short a time interval or insensitive to that parameter.
This is practical non-identifiability.
Structural versus practical
| Structural identifiability | Practical identifiability |
|---|---|
| ideal noise-free information | actual finite noisy data |
| mathematical property of model plus observations | depends strongly on dataset and experimental design |
| cannot be fixed merely by measuring more precisely | may improve with better or more informative data |
Objective-function valleys
Suppose parameters are estimated by minimising
\[J(\boldsymbol\theta).\]If many parameter combinations give nearly the same value of \(J\), the objective surface can contain a long shallow valley.
An optimiser may return one point in the valley even though many other points fit almost equally well.
Likelihood ridges
The same phenomenon appears in likelihood inference. A ridge occurs when
\[\ell(\boldsymbol\theta)\]changes very little along some parameter direction.
A numerical maximum on a flat ridge should not be interpreted as strong evidence for one precise parameter vector.
Parameter correlation
If increasing one parameter can be compensated by decreasing another, fitted parameters may be strongly correlated.
For two estimates, this can be reflected by
\[\operatorname{Corr}(\hat\theta_1,\hat\theta_2)\approx\pm1.\]Strong correlation is a useful warning sign, although correlation alone does not establish structural non-identifiability.
Sensitivity and identifiability
A local sensitivity of an observable to parameter \(\theta_j\) is
\[\boxed{s_j(t)=\frac{\partial y(t;\boldsymbol\theta)}{\partial\theta_j}}.\]If this sensitivity is very small throughout the observed region, the data may contain little information about that parameter.
If two parameters have nearly proportional sensitivity patterns, their separate effects may be difficult to distinguish.
Sensitivity matrix
For observations at times \(t_i\), define
\[S_{ij}=\frac{\partial\mu_i}{\partial\theta_j}.\]The columns describe how each parameter changes the predicted observations.
Nearly dependent columns indicate that different parameter changes produce similar effects on the data.
Rank and local information
If the sensitivity matrix lacks full column rank, some local parameter directions cannot be distinguished from the observations.
Full local rank is useful evidence of local distinguishability under appropriate conditions, but it is not by itself a complete global structural-identifiability proof.
Fisher information
For independent Gaussian observations with variance \(\sigma^2\), a local information matrix has the form
\[\boxed{F=\frac{1}{\sigma^2}S^TS}.\]If \(F\) is singular, at least one local parameter direction contains no independent information under this approximation.
If \(F\) is nearly singular, estimates can be extremely uncertain.
Eigenvalues of the information matrix
Small eigenvalues of \(F\) correspond to weakly informed combinations of parameters.
A very large ratio between the largest and smallest eigenvalues indicates an ill-conditioned estimation problem.
This is a local diagnostic rather than a universal proof of identifiability.
Confidence intervals as a warning
Very wide confidence intervals often indicate weak practical identifiability.
An effectively unbounded profile-likelihood interval is especially informative because it shows that the data do not strongly constrain the parameter in at least one direction.
Profile likelihood
Fix one parameter \(\theta_j\) and re-optimise the others to obtain
\[\ell_p(\theta_j).\]A sharply peaked profile indicates stronger practical information, while a broad or flat profile indicates weak determination.
Profile likelihood can reveal asymmetry and parameter compensation hidden by standard-error approximations.
Multiple starting values
Running an optimiser from many initial parameter guesses can expose non-identifiability or multimodality.
If very different fitted parameters repeatedly produce almost identical objective values and trajectories, the inference problem requires further investigation.
However, agreement between starting values does not prove structural identifiability.
More data are not always enough
If two parameters affect the observable only through the same combination, collecting more measurements of that same observable cannot separate them.
Structural non-identifiability requires new kinds of information rather than merely more of the same data.
More informative measurements
Measuring an additional state variable can break a parameter confounding relationship.
For example, observing both incidence and prevalence may distinguish mechanisms that are difficult to separate using only one of them.
More informative times
Practical identifiability can depend strongly on when measurements are collected.
Early growth, peak behaviour and decline may contain information about different parameters. Concentrating every observation in one nearly steady region can leave dynamic parameters weakly informed.
Experimental interventions
Changing an experimental condition can reveal parameter effects that remain hidden under a single condition.
Examples include different doses, initial populations, environmental conditions or controlled interventions.
Experimental design can therefore be used deliberately to improve identifiability.
Fixing parameters
If a parameter is accurately known from independent evidence, fixing it can reduce confounding and make remaining parameters more estimable.
But fixing an uncertain parameter at an arbitrary value hides uncertainty rather than solving it.
Using external information
Independent experiments, literature estimates or additional datasets can provide information unavailable from the primary dataset.
The source and uncertainty of this external information should be represented transparently.
Reparameterisation
If only a combination such as
\[\phi=\frac{\beta}{\gamma}\]is well determined, it may be scientifically preferable to parameterise or report the model in terms of \(\phi\) rather than claim precise separate estimates of \(\beta\) and \(\gamma\).
Model simplification
Removing unsupported complexity can improve interpretability and practical estimation.
A simpler model with identifiable parameters can be more scientifically useful than a more detailed model whose parameters cannot be distinguished by the available observations.
Identifiability versus observability
Observability traditionally asks whether the internal state of a dynamic system can be reconstructed from its outputs, while identifiability asks whether unknown parameters can be determined.
The concepts are related because unknown parameters can sometimes be treated as additional constant states, but they are not identical questions.
Identifiability versus sensitivity
Sensitivity asks how much an output changes when a parameter changes.
Identifiability asks whether parameter effects can be uniquely distinguished from each other using observations.
A parameter can influence the output strongly yet still be non-identifiable if another parameter produces the same effect.
Identifiability versus uncertainty
Uncertainty quantification describes the range of plausible parameter or prediction values under a specified inferential framework.
Identifiability helps explain why that uncertainty may be large, highly correlated or unbounded.
Identifiability before biological interpretation
Suppose an optimiser returns
\[\hat\beta=0.31,\qquad\hat\gamma=0.12.\]Reporting biological conclusions from these values is unsafe if many substantially different pairs fit the observations equally well.
The fitted trajectory may be well determined while the mechanistic parameters are not.
A practical diagnostic workflow
First determine whether the model and observation scheme are structurally identifiable. Then fit the model and inspect objective or likelihood surfaces, parameter correlations, profile likelihoods, confidence intervals and sensitivity information. Finally, test whether additional measurements or experimental conditions improve parameter determination.
Transition to sensitivity analysis
Identifiability depends partly on how model outputs respond to changes in parameters. The next lesson develops sensitivity analysis directly: measuring which parameters most strongly influence states, outputs and biologically important predictions.