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.

Identifiability from linear observations asks which features of an unknown vector $\theta \in {\mathbb{R}}^p$ are fixed by observing $y=A\theta$. Two parameter vectors are observationally equivalent exactly when their difference lies in the kernel of a linear map $A$:Thus the full vector is identifiable precisely when $\ker A=\{0\}$, equivalently when $A$ has full column matrix rank. Repeating the same rows can improve precision in a noisy measurement without increasing this rank.

A linear target $g(\theta )=c^{\mathsf{T}}\theta$ requires less information. It is identifiable if and only if $c$ is in the row space of $A$. If $c=A^{\mathsf{T}}d$, then $g(\theta )=d^{\mathsf{T}}y$, so the target is recovered directly. Conversely, an identifiable target must vanish on every observationally invisible difference: $c^{\mathsf{T}}h=0$ for every $h\in \ker A$. Orthogonality to the kernel is exactly row-space membership. This is a concrete instance of quantity-specific identifiability; the primary identification literature also treats identification of individual parameter components by rank conditions.^[ref]

For an original example, supposeBoth $(1,4,3)$ and $(2,3,4)$ fit perfectly. Their difference is a multiple of $(1,-1,1)$, which spans the kernel. The target ${\theta }_1-{\theta }_3=y_1-y_2=-2$ is identified, whereas ${\theta }_1+{\theta }_2+{\theta }_3$ takes values 8 and 9 and is not. A new independent measurement of ${\theta }_2$ restores full identification; another copy of $y_1$ does not.

The same distinction applies to a statistical model $Y\sim N(A\theta ,\Sigma )$ with known positive-definite covariance matrix $\Sigma$: two parameters give the same observable law precisely when their means agree. A Moore-Penrose pseudoinverse selects one solution when the system is underdetermined, but that numerical convention does not show that the selected vector is the unique physical parameter. Additional restrictions must be justified as assumptions, not silently supplied by a solver.