Identifiability

In statistics, identifiability is a property which a model must satisfy in order for inference to be possible. We say that the model is identifiable if it is theoretically possible to learn the true value of this model’s underlying parameter after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameter must generate different probability distributions of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the identification conditions.

In some cases, the model is unidentifiable, but it is still possible to learn the true values of a certain subset of the model parameters. In this case we say that the model is partially identifiable. In other cases it may be possible to learn the location of the true parameter up to a certain finite region of the parameter space, in which case the model is set identifiable.

Definition

Let ℘ = {P_θ: θ∈Θ} be a statistical model where the parameter space Θ is either finite- or infinite-dimensional. We say that ℘ is identifiable if the mapping θ ↦ P_θ is one-to-one:^[1]

$P_{\theta_1}=P_{\theta_2} \quad\Rightarrow\quad \theta_1=\theta_2 \quad\ \text{for all } \theta_1,\theta_2\in\Theta.$

This definition means that distinct values of θ should correspond to distinct probability distributions: if θ₁≠θ₂, then also P_θ1≠P_θ2.^[2] If the distributions are defined in terms of the probability density functions, then two pdfs should be considered distinct only if they differ on a set of non-zero measure (for example two functions ƒ₁(x)=1_0≤x<1 and ƒ₂(x)=1_0≤x≤1 differ only at a single point x=1 — a set of measure zero — and thus cannot be considered as distinct pdfs).

Identifiability of the model in the sense of invertibility of the map θ ↦ P_θ is equivalent to being able to learn the model’s true parameter if the model can be observed indefinitely long. Indeed, if {X_t}⊆S is the sequence of observations from the model, then by the strong law of large numbers,

$\frac{1}{T} \sum_{t=1}^T \mathbf{1}_{\{X_t\in A\}} \ \xrightarrow{a.s.}\ \operatorname{Pr}[X_t\in A],$

for every measurable set A⊆S (here 1_{…} is the indicator function). Thus with an infinite number of observations we will be able to find the true probability distribution P₀ in the model, and since the identifiability condition above requires that the map θ ↦ P_θ be invertible, we will also be able to find the true value of the parameter which generated given distribution P₀.

Examples

Example 1

Let ℘ be the normal location-scale family:

$\mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} e^{ -\frac{1}{2\sigma^2}(x-\mu)^2 }\ \Big|\ \theta=(\mu,\sigma): \mu\in\mathbb{R}, \,\sigma\!>0 \ \Big\}.$

Then

$\begin{align} f_{\theta_1}=f_{\theta_2}\ &\Leftrightarrow\ \tfrac{1}{\sqrt{2\pi}\sigma_1}e^{ -\frac{1}{2\sigma_1^2}(x-\mu_1)^2 } = \tfrac{1}{\sqrt{2\pi}\sigma_2}e^{ -\frac{1}{2\sigma_2^2}(x-\mu_2)^2 } \\ &\Leftrightarrow\ \tfrac{1}{\sigma_1^2}(x-\mu_1)^2 + \ln \sigma_1^2 = \tfrac{1}{\sigma_2^2}(x-\mu_2)^2 + \ln \sigma_2^2 \\ &\Leftrightarrow\ x^2\big(\tfrac{1}{\sigma_1^2}-\tfrac{1}{\sigma_2^2}\big) - 2x\big(\tfrac{\mu_1}{\sigma_1^2}-\tfrac{\mu_2}{\sigma_2^2}\big) + \big(\tfrac{\mu_1^2}{\sigma_1^2}-\tfrac{\mu_2^2}{\sigma_2^2}+\ln\sigma_1^2-\ln\sigma_2^2\big) = 0 \end{align}$

This expression is equal to zero for almost all x only when all its coefficients are equal to zero, which is only possible when |σ₁| = |σ₂| and μ₁ = μ₂. Since in the scale parameter σ is restricted to be greater than zero, we conclude that the model is identifiable: ƒ_θ1=ƒ_θ2 ⇔ θ₁=θ₂.

Example 2

Let ℘ be the standard linear regression model:

$y = \beta'x + \varepsilon, \quad \operatorname{E}[\,\varepsilon|x\,]=0$

(where ′ denotes matrix transpose). Then the parameter β is identifiable if and only if the matrix E[xx′] is invertible. Thus, this is the identification condition in the model.

Example 3

Suppose ℘ is the classical errors-in-variables linear model:

$\begin{cases} y = \beta x^* + \varepsilon, \\ x = x^* + \eta, \end{cases}$

where (ε,η,x*) are jointly normal independent random variables with zero expected value and unknown variances, and only the variables (x,y) are observed. Then this model is not identifiable,^[3] only the product βσ²_∗ is (where σ²_∗ is the variance of the latent regressor x*). This is also an example of set identifiable model: although the exact value of β cannot be learned, we can guarantee that it must lie somewhere in the interval (β_yx, 1÷β_xy), where β_yx is the coefficient in OLS regression of y on x, and β_xy is the coefficient in OLS regression of x on y.^[4]

If we abandon the normality assumption and require that x* were not normally distributed, retaining only the independence condition ε⊥η⊥x*, then the model becomes identifiable.^[3]

Software

In the case of parameter estimation in partially observed dynamical systems, the profile likelihood can be also used for structural and practical identifiability analysis^[5]. An implementation of the Profile Likelihood Approach is available in the MATLAB Toolbox PottersWheel.

See also

• Parameter identification problem

References

• Casella, George; Roger L. Berger (2001), Statistical inference (2nd ed.), ISBN 0-534-24312-6 
• Hsiao, Cheng (1983), Identification, Handbook of Econometrics, Vol. 1, Ch.4, North-Holland Publishing Company 
• Lehmann, E. L.; G. Casella (1998), Theory of point estimation (2nd ed.), Springer, ISBN 0-387-98502-6 
• Reiersøl, Olav (1950), "Identifiability of a linear relation between variables which are subject to error", Econometrica (The Econometric Society) 18 (4): 375–389, doi:10.2307/1907835, JSTOR 1907835 
• van der Vaart, A.W. (1998), Asymptotic Statistics, Cambridge University Press, ISBN 978-0-521-49603-2 

Notes

1. ^ Lehmann & Casella 1998, Definition 1.5.2
2. ^ van der Vaart 1998, p. 62
3. ^ ^{a b} Reiersøl 1950
4. ^ Casella & Berger 2001, p. 583
5. ^ Raue, A; Kreutz, C; Maiwald, T; Bachmann, J; Schilling, M; Klingmüller, U; Timmer, J (2009). "Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood". Bioinformatics 25 (15): 1923–9. doi:10.1093/bioinformatics/btp358. PMID 19505944. http://bioinformatics.oxfordjournals.org/cgi/content/abstract/btp358?ijkey=iYp4jPP50F5vdX0&keytype=ref.