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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.

Nonregular identification at a flat mean map occurs when different parameter values give different probability distributions, yet the first derivative of the observable mean vanishes at a parameter of interest. Identifiability is a property of the entire parameter-to-distribution map; a first-order sensitivity calculation can miss it.

Consider an original statistical model with independent observationsThe cubic mean map is a strictly increasing function, so equality of the normal distributions implies equality of $\theta$. The model is globally identifiable, including at $\theta =0$. Nevertheless, the single-observation score function and Fisher information areThus $I(0)=0$ does not imply nonidentifiability. The rank is zero at the origin and one at every nonzero nearby point. This is precisely why a regular-point assumption matters: Rothenberg's local identification criterion assumes constant information-matrix rank in a neighborhood.^[ref]

Maximum likelihood estimation gives $\hat{\theta }=\mathrm{cbrt}(\overline{Y})$, using the real cube root. At the true value zero, write $\overline{Y}=Z/\sqrt{n}$ with a standard normal random variable $Z$. ThenThis is an exact sampling distribution identity for every $n$, not merely a convergence in distribution statement. The estimation error has scale $n^{-1/6}$ and a nonnormal shape. At a fixed nonzero $\theta$, the usual delta method instead gives asymptotic variance $1/(9n{\theta }^4)$; that approximation cannot be extended uniformly through zero.

An exact confidence interval through a cubic mean map follows by applying the strictly increasing function $\mathrm{cbrt}$ to both endpoints of the ordinary known-variance confidence interval for the mean:Here $z_p$ is the quantile function of the standard normal distribution. The transformed interval has exact coverage $1-\alpha$ for every $\theta$. When $n=64$ and the observed sample mean is zero, the 95% interval is approximately $[-0.626,0.626]$. Multiplying $n$ by 64 halves its width at that observed mean. A first-order interval based on dividing by the zero mean-map derivative would fail instead.

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.

Thomas J. Rothenberg is a mathematical econometrician whose work concerns statistical inference and the identifiability of models. The American Economic Association's 2015 citation recognizes his contributions to identification and inference and identifies him as a professor emeritus at Berkeley.^[ref]

His primary paper Identification in Parametric Models by Thomas J. Rothenberg, published in 1971, develops identification criteria beyond one model considered in isolation. It connects local identification with the information matrix under regularity conditions and treats identification of individual parameters separately from that of the full vector.^[ref] The distinction is useful when a scientifically relevant combination is recoverable even though the entire parameterization is not; identifiability from linear observations gives an explicit example.

An original counterexample explains why the regularity qualifications matter. Consider one observation with normal distributionThe mean map is one-to-one, so distinct parameters have distinct observable distributions. Nevertheless its Fisher information iswhich vanishes at zero. The information rank changes there, so a nonsingularity criterion cannot be applied without its assumptions. Nonregular identification at a flat mean map separates exact distinguishability from first-order sensitivity: a zero derivative does not by itself establish that two different parameter values produce the same law.

The example illustrates a broader contribution of identification theory. Before asking an estimator for a precise answer, determine whether the observations can distinguish the proposed quantities and whether the local approximation used for inference is valid.