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Non-identifiability of shape from height and mass means that different external geometries can produce the same measured height and mass. The observation map from a body shape to those two numbers is many-to-one. A formula using only those inputs cannot distinguish shapes that the observations themselves do not distinguish. This is an identifiability issue in an inverse problem, independently of measurement precision.

An original family of homogeneous cuboids supplies an explicit proof. Fix height $H$, mass $M$ and density $\rho$, and define horizontal area $A=M/(\rho H)$. LetEvery real $t$ gives $ab=A$, the same height and the same mass. Its external surface area, however, isThe square section $t=0$ minimizes this family's area. At $t=\log 2$, the sides have ratio four and the lateral surface area is 25% larger. Height and mass remain exactly unchanged. This is a geometric construction, not a proposed anatomical model of a person.

With stipulated $H=1.8$ m, $M=80$ kg and $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, those two total areas are approximately $1.607$ and $1.986{\mathrm{m}}^2$. The Mosteller formula assigns both the same predicted body surface area, $2.00{\mathrm{m}}^2$, because its documented inputs are only height and mass.^[ref] No reduction of numerical rounding can make one identical prediction equal both different geometric areas.

Additional measurements and anatomical identifiability can change the problem. In this particular cuboid family, measuring the horizontal perimeter $P=2(a+b)$ identifies the area through $S=2A+HP$. That conclusion relies on the assumed geometry; a perimeter does not generally determine the full external surface of an arbitrary human body.

Validation must therefore ask whether the observations and model contain the shape information needed for the intended quantity. Repeated precise height and mass measurements constrain the inputs. Independent geometric measurements or a validated restriction on allowable shapes constrain the missing anatomical information.

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.