Linearized propagation

Use this page when uncertainties are small enough that a local derivative may summarize how a smooth scientific map changes their covariance.

The scientific question¶

Given an input vector with a stated mean and covariance, how does a smooth map change those moments near a chosen expansion point? Linearized propagation answers that local question efficiently. It is useful for sensitivity diagnosis and for small, approximately unimodal perturbations, but it is not an exact general propagation rule.

The expansion point matters. A Jacobian evaluated at the mean answers a different local question from a Jacobian evaluated at a nominal best fit or another reference state. The caller must record that point and the units and ordering of every input and output component.

Mathematical objects¶

Let $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}$ be differentiable near an arbitrary expansion point $\mathbf{x}_{0}$. Let $\mathbf{X}$ have mean $\boldsymbol{\mu}_{x}=\mathbb{E}[\mathbf{X}]$ and covariance $\mathbf{C}_{x}\in\mathbb{R}^{d\times d}$. Define $\mathbf{Y}=f(\mathbf{X})$ and the Jacobian $\mathbf{J}_{0}\in\mathbb{R}^{m\times d}$ evaluated at $\mathbf{x}_{0}$ by $(J_0)_{ai}=\partial f_a/\partial x_i$. The expansion point may equal the mean, but the derivation does not require that choice.

The units expose the contract. $(J_0)_{ai}$ has units $[Y_a]/[X_i]$, $(C_x)_{ij}$ has units $[X_i][X_j]$, and $(C_y)_{ab}$ must have units $[Y_a][Y_b]$. Off-diagonal entries are cross-covariances, not optional noise terms. Dropping them changes both the input model and the propagated answer.

Rank is also informative. If $\mathbf{C}_{x}$ has rank $r$, the first-order output covariance has rank at most $\min(r,\operatorname{rank}\mathbf{J}_{0})$. A singular covariance can represent exact constraints or an under-resolved model; adding a small diagonal floor changes that scientific statement.

Core derivation¶

At the arbitrary point $\mathbf{x}_{0}$, a first-order Taylor expansion gives

Taking the expectation of (1) defines the approximate output mean. Subtracting that mean centers the linearized output about $\boldsymbol{\mu}_{x}$ rather than about the expansion point:

The expected outer product of the centered relation in (2) is therefore

Equation (3) follows because $\mathbb{E}[(\mathbf{X}-\boldsymbol{\mu}_{x}) (\mathbf{X}-\boldsymbol{\mu}_{x})^{\mathsf{T}}]=\mathbf{C}_{x}$. By contrast, the raw second moment around an arbitrary expansion point is

Thus (4) equals $\mathbf{C}_{x}$ only when the expansion point equals the input mean. For an affine map $f(\mathbf{x})=\mathbf{A}\mathbf{x}+\mathbf{b}$, the equation is exact with $\mathbf{J}_{0}=\mathbf{A}$ whenever the second moments exist. For a nonlinear map, neglected Hessian and higher-order terms can shift both the mean and the covariance.

If inputs $\mathbf{X}$ and $\mathbf{Z}$ are jointly uncertain, their full block covariance includes $\mathbf{C}_{xz}$. Propagating $f(\mathbf{X},\mathbf{Z})$ requires the augmented Jacobian and those cross-covariances. Propagating the blocks separately silently assumes independence.

Failure modes and interpretation limits¶

• Strong curvature makes a single tangent map unrepresentative across the input support and can bias both mean and covariance.

• Discontinuities, clipping, branch changes, thresholds, and discrete indices do not have one scientifically meaningful local derivative at their boundary.

• Multimodal inputs can share a covariance while mapping to separated output modes that no covariance ellipse describes.

• Ill-conditioned $\mathbf{J}_{0}$ can amplify small input or floating-point changes; a finite output is not evidence of a stable calculation.

• Rank deficiency can be physical, structural, or numerical. Regularization must state which interpretation it changes.

• A Jacobian at an arbitrary nominal point is not automatically a Jacobian at the expectation, posterior mode, or true state.

• A local AD result differentiates the executed program, including any smoothing, clipping, or branch semantics present in that program.

What Jaxstro may add¶

JAX owns transformations and random primitives. NumPyro and BlackJAX own probabilistic inference and sampling mechanics. Informax owns inference-aware scientific workflows in the Jaxstro ecosystem. A future jaxstro.uncertainty would own only domain-agnostic propagation representations, unit and shape conventions, deterministic key policy, provenance, and evidence contracts. Covariance propagation does not perform inference and does not validate a probability model.

A future helper may compute Jacobian-vector or vector-Jacobian factorizations, preserve component and unit metadata, and report symmetry, rank, and conditioning diagnostics. It may not choose an inference model, insert undocumented covariance floors, discard cross-covariances, or call the approximation exact.

Evidence required before implementation¶

The minimum evidence includes analytic affine maps, independently calculated Jacobians, unit-consistency tests, symmetry and positive-semidefinite checks, rank-deficient examples, and central finite-difference comparisons on a smooth domain. Nonlinear cases must compare the first-order result with high-resolution ensemble propagation as input scale shrinks, demonstrating the expected local convergence rather than agreement at one hand-picked scale.

Tests must cover correlated inputs, rectangular Jacobians, batched covariances, JIT and VMAP behavior, float32 and float64 conditioning, and explicit failures for nonfinite values or mismatched shapes. Evidence should report the expansion point and must separate numerical agreement from validation of the probability model.

Claim boundary¶

The formula on this page is exact for affine maps and a first-order approximation for general smooth maps. It does not reconstruct tails, modes, support, or a posterior; it does not justify independence assumptions; and it does not certify that the input covariance is scientifically meaningful. No runtime surface exists, and no implementation or release date is implied.

Connected representations, foundations, and methods¶

• Start with What uncertainty represents to identify the uncertainty being propagated.

• Return to Scientific representations for the larger representation boundary.

• Review What is a derivative? and Sensitivity, conditioning, and identifiability for Jacobians and conditioning.

• Compare Autodiff products for executed derivative semantics and Ensemble propagation for a nonlocal alternative.