 Quantity-specific identifiability

Quantity-specific identifiability asks whether a chosen function of a model's parameters is determined by the observable probability distribution, even when the complete parameter vector is not. For model $\{P_{\theta }\}$ and target $g(\theta )$, the criterion isEquivalently, the target must be constant on each class of observationally equivalent parameters. This separates identification of a scientifically meaningful quantity from reconstruction of every internal parameter. Thomas J. Rothenberg's primary treatment explicitly distinguishes identification of individual parameter components from that of a full system.^[ref]

For an original example, let observations have a Poisson distribution with mean $a+b$, where $a,b>0$. Parameter pairs $(a,b)=(2,3)$ and $(1,4)$ give exactly the same observable law. The sum $g(a,b)=a+b$ is identifiable, but $a/(a+b)$ is not: its values are $0.4$ and $0.2$. With $n$ independent observations, the sample mean estimates the sum with variance $(a+b)/n$. More observations can improve that estimate without distinguishing the two allocations.

Identifiability from linear observations gives a useful exact criterion. Suppose the observation law identifies $A\theta$ and contains no additional information about $\theta \in {\mathbb{R}}^p$. A linear target $c^{\mathsf{T}}\theta$ is identifiable exactly when $c$ belongs to the row space of $A$. To prove this, observationally equivalent parameter differences lie in the kernel of a linear map $A$. The target is unchanged for every such difference precisely when $c$ is orthogonal to that kernel, which is equivalent to membership in the row space. If $c=A^{\mathsf{T}}d$, the target can be recovered as $d^{\mathsf{T}}(A\theta )$.

A deterministic geometric analogue appears in additional measurements and anatomical identifiability: measured perimeter can determine the surface area of the stipulated cuboid family even when exchanging its two horizontal side labels remains invisible. The target and the allowed model class must be stated.

Identification is not the same as good precision. A formally identifiable target can still be weakly constrained by finite noisy data. Conversely, a nonidentifiable target requires additional assumptions or observations, rather than merely more repeated data from the same indistinguishable law.