Identifiability
In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameters 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.
A model that fails to be identifiable is said to be non-identifiable or unidentifiable: two or more parametrizations are observationally equivalent. In some cases, even though a model is non-identifiable, 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.
Aside from strictly theoretical exploration of the model properties, identifiability can be referred to in a wider scope when a model is tested with experimental data sets, using identifiability analysis.[1]
Definition[edit]
Let be a statistical model with parameter space . We say that is identifiable if the mapping is one-to-one:[2]
This definition means that distinct values of
Identifiability of the model in the sense of invertibility of the map is equivalent to being able to learn the model's true parameter if the model can be observed indefinitely long. Indeed, if {Xt} ⊆ S is the sequence of observations from the model, then by the strong law of large numbers,
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 be invertible, we will also be able to find the true value of the parameter which generated given distribution P0.
Examples[edit]
Example 1[edit]
Let be the normal location-scale family:
Then
This expression is equal to zero for almost all x only when all its coefficients are equal to zero, which is only possible when |
Example 2[edit]
Let be the standard linear regression model:
(where ′ denotes matrix transpose). Then the parameter
Example 3[edit]
Suppose is the classical errors-in-variables linear model:
where (
If we abandon the normality assumption and require that x* were not normally distributed, retaining only the independence condition
See also[edit]
References[edit]
Citations[edit]
- ^ Raue, A.; Kreutz, C.; Maiwald, T.; Bachmann, J.; Schilling, M.; Klingmuller, U.; Timmer, J. (2009-08-01). "Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood". Bioinformatics. 25 (15): 1923–1929. doi:10.1093/bioinformatics/btp358. PMID 19505944.
- ^ Lehmann & Casella 1998, Ch. 1, Definition 5.2
- ^ van der Vaart 1998, p. 62
- ^ a b Reiersøl 1950
- ^ Casella & Berger 2002, p. 583
Sources[edit]
- Casella, George; Berger, Roger L. (2002), Statistical Inference (2nd ed.), ISBN 0-534-24312-6, LCCN 2001025794
- Hsiao, Cheng (1983), Identification, Handbook of Econometrics, Vol. 1, Ch.4, North-Holland Publishing Company
- Lehmann, E. L.; Casella, G. (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, 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
Further reading[edit]
- Walter, É.; Pronzato, L. (1997), Identification of Parametric Models from Experimental Data, Springer
Econometrics[edit]
- Lewbel, Arthur (2019-12-01). "The Identification Zoo: Meanings of Identification in Econometrics". Journal of Economic Literature. 57 (4). American Economic Association: 835–903. doi:10.1257/jel.20181361. ISSN 0022-0515. S2CID 125792293.
- Matzkin, Rosa L. (2013). "Nonparametric Identification in Structural Economic Models". Annual Review of Economics. 5 (1): 457–486. doi:10.1146/annurev-economics-082912-110231.
- Rothenberg, Thomas J. (1971). "Identification in Parametric Models". Econometrica. 39 (3): 577–591. doi:10.2307/1913267. ISSN 0012-9682. JSTOR 1913267.