In statistics, identifiability is a property which a model must satisfy for precise inference to be possible.
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.
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.
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.
Since in the scale parameter σ is restricted to be greater than zero, we conclude that the model is identifiable: ƒθ1 = ƒθ2 ⇔ θ1 = θ2.
be the standard linear regression model: (where ′ denotes matrix transpose).
is the classical errors-in-variables linear model: 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.
This is also an example of a set identifiable model: although the exact value of β cannot be learned, we can guarantee that it must lie somewhere in the interval (βyx, 1÷βxy), where βyx is the coefficient in OLS regression of y on x, and βxy is the coefficient in OLS regression of x on y.