Chain-Rule Error Propagation In Nested Models

Why nested models need the chain rule

A measured quantity often passes through several calculations before it becomes the output of a model. A sensor reading might first be converted into an intermediate quantity, then used in another formula. Even if each stage is simple, an error in the original measurement can affect every later stage.

The chain rule provides a direct way to track this effect. It tells us how the output changes with the original input by multiplying the derivatives of the stages. The resulting derivative is a local sensitivity: it estimates the output change per unit change in the input near the measured value.

For a small measurement uncertainty, this sensitivity gives a first-order estimate of the corresponding output uncertainty. It is an estimate based on linearization, not an exact guarantee about finite changes.

Nested models and propagated uncertainty

Suppose an input \(x\) passes through an intermediate stage \(u=g(x)\), followed by an output stage \(y=h(u)\). The complete model is the composition \(y=h(g(x))\). Assume \(g\) is differentiable at the input value \(x_0\), and \(h\) is differentiable at the intermediate value \(u_0=g(x_0)\).

The absolute value is used when estimating the size of a worst-case output change from a bounded input uncertainty. For a signed predicted change \(dx\), keep the sign instead: \(dy=h'(g(x_0))g'(x_0)\,dx\). Thus the derivative gives both direction and scale, while the absolute value gives a nonnegative uncertainty magnitude.

For more than two stages, multiply one derivative for each stage. Each derivative must be evaluated at the input value actually received by that stage—not automatically at the original measurement.

Why the sensitivities multiply

The multiplication follows from how small changes pass from one stage to the next. For a small input change \(dx\), the first stage predicts an intermediate change \(du \approx g'(x_0)\,dx\). The second stage then predicts \(dy \approx h'(u_0)\,du\). Substituting the first estimate into the second gives \(dy \approx h'(u_0)g'(x_0)\,dx\).

This argument describes the first-order estimate. The derivative result itself can be justified by tracking the remainders in the differentiability statements. Write \(\Delta x\) for a small finite input change and \(\Delta u=g(x_0+\Delta x)-g(x_0)\). Differentiability of \(g\) gives

Here \(o(\Delta x)\) denotes a remainder which, divided by \(\Delta x\), tends to zero as \(\Delta x\) tends to zero. In particular, \(\Delta u=O(\Delta x)\): the intermediate change is at most of first order in the input change near \(x_0\). Differentiability of \(h\) at \(u_0=g(x_0)\) gives

Substitute the expression for \(\Delta u\). Since \(\Delta u=O(\Delta x)\), the remainder \(o(\Delta u)\) is also \(o(\Delta x)\). Therefore

Dividing by \(\Delta x\) and taking the limit as \(\Delta x\) tends to zero proves that the derivative of the composition at \(x_0\) is \(h'(g(x_0))g'(x_0)\). The product is not a convenient shortcut unrelated to the model: it records the two successive changes, from \(x\) to \(u\) and then from \(u\) to \(y\).

Worked Example: A logarithm of a scaled square

Worked Example: A trigonometric stage inside a square root

Worked Example: An exponential of a reciprocal

A common mistake: skipping an intermediate derivative

For \(y=\sqrt{5+\sin\theta}\) at \(\theta=0\), one might incorrectly use only the derivative of the square-root stage and write \(dy/d\theta=1/(2\sqrt{5})\) without accounting for how \(\sin\theta\) changes with \(\theta\). That calculation happens to give the correct number at zero because \(d(5+\sin\theta)/d\theta=\cos 0=1\). It is not a valid general rule.

At \(\theta=\pi/3\), the intermediate value is \(u=5+\sqrt{3}/2\), and the inner derivative is \(\cos(\pi/3)=1/2\). The correct sensitivity is \((1/2)(1/[2\sqrt{5+\sqrt{3}/2}])\); omitting the inner derivative gives twice this value. The correction is to identify every stage, evaluate its derivative at the stage’s input, and multiply all of them.

Interpreting the result

Chain-rule error propagation is useful whenever a measurement is transformed through a sequence of formulas. The total derivative compresses the local effect of all those stages into one sensitivity at the measured input. A large absolute sensitivity means a small input uncertainty can produce a larger output change; a small one means the model locally dampens that change.

The estimate is most informative when the uncertainty is small enough that the model is well approximated by its tangent near the measured value. It does not by itself give an exact bound for a finite input interval. The sign of the derivative describes the direction of the predicted change, while the absolute value describes its estimated magnitude. No probability model is assumed: this is a first-order propagation of a bounded measurement uncertainty, not a statistical combination of random errors.