What uncertainty represents

Use this page when a result is called “uncertain” and you need to identify which source of incomplete knowledge is actually being represented before choosing a propagation method.

The scientific question¶

What does a reported distribution, covariance, interval, or ensemble say about a scientific quantity, and what does it leave unsaid? The answer must identify the random object, its physical support and units, the conditioning information, and the process that generated the representation. Without those details, propagating numbers through a function can produce a precise-looking output with no stable scientific interpretation.

At least five categories must remain distinct:

Variation represented by a probability model describes variation across a defined population or repeated process.
Uncertainty about parameters or latent state describes incomplete knowledge about a particular system, conditional on a model and observations.
Measurement and noise models describe how an instrument or data-generating process connects latent quantities to recorded values.
Numerical approximation error arises from discretization, iteration, truncation, finite sampling, or finite precision.
Model discrepancy represents systematic inadequacy between the scientific model and the process being studied.

These categories are not interchangeable. Every representation must state units and support explicitly. More Monte Carlo draws reduce sampling error but do not repair model discrepancy. A wider measurement-noise distribution does not automatically represent uncertainty in a calibration parameter.

Mathematical objects¶

Let the random vector $\mathbf{X}$ take values in a stated support $\mathcal{X}$ with probability law $P_{\mathbf{X}}$ . Its components have units; therefore the mean $\boldsymbol{\mu}_{X}$ has the units of $\mathbf{X}$ , while the covariance entry $(\mathbf{C}_{X})_{ij}$ has units $[X_i][X_j]$ . A covariance between length and velocity, for example, has units of length times velocity.

Support is part of the contract. A positive mass, an angle on a circle, a bounded fraction, and an unconstrained real parameter are different mathematical objects even if each is stored as a floating-point scalar. Conditioning is also part of the object: $P(\mathbf{X}\mid D,M)$ depends on data $D$ and model $M$ and is not the same claim as a population distribution $P(\mathbf{X})$ .

A covariance matrix is not a complete probability distribution. It does not by itself specify support, tail behavior, skewness, multimodality, or the distinction between probability mass and probability density. Two laws can share the same mean and covariance while assigning very different probability to a scientific event.

Core derivation¶

For a random vector with finite second moments, mean and covariance are summaries of a probability law:

\boldsymbol{\mu}_{X}=\mathbb{E}[\mathbf{X}], \qquad \mathbf{C}_{X}=\mathbb{E}\!\left[ (\mathbf{X}-\boldsymbol{\mu}_{X}) (\mathbf{X}-\boldsymbol{\mu}_{X})^{\mathsf{T}} \right].

(1)

Equation (1) follows by defining the centered variable $\boldsymbol{\eta}=\mathbf{X}-\boldsymbol{\mu}_{X}$ and taking the expected outer product $\mathbb{E}[\boldsymbol{\eta}\boldsymbol{\eta}^{\mathsf{T}}]$ . The diagonal records marginal variances; the off-diagonal entries record linear cross-covariances. The derivation requires finite second moments and says nothing about Gaussianity. It is a summary operation, not an inference algorithm.

When $\mathbf{Y}=f(\mathbf{X})$ , the output law is the pushforward $P_{\mathbf{Y}}=f_{\#}P_{\mathbf{X}}$ . Linearization, sigma points, and ensembles are different approximations or numerical representations of that pushforward. They cannot determine whether the input law was scientifically warranted.

Failure modes and interpretation limits¶

A covariance-only description can conceal skewness, heavy tails, disconnected support, or multiple modes.
Treating numerical approximation error as independent random noise can hide a deterministic bias or convergence failure.
Combining measurement error and model discrepancy without a generative model can make their contributions non-identifiable.
Mixing components with unrecorded units can make covariance addition meaningless.
Ignoring support can assign probability to impossible states.
Calling posterior uncertainty “population variation” changes the conditioning claim and the scientific question.
A positive-semidefinite covariance is necessary for a second-moment summary, but it does not validate the underlying probability model.

These are representation failures, not problems that automatic differentiation or larger sample counts can resolve.

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.

Such a surface may define explicit records for moment summaries, keyed ensembles, and propagation diagnostics. It must require the caller to state component order, shape, units, support, and provenance. It must not construct posteriors, choose likelihoods, or decide which uncertainty category a domain workflow should use.

Evidence required before implementation¶

Implementation would require analytic pushforward cases, unit and shape checks, positive-semidefinite diagnostics, singular and rank-deficient cases, deterministic key replay, and comparisons among linearized, sigma-point, and ensemble results on controlled nonlinear maps. Tests must show that category and provenance metadata survive JAX transformations and serialization. Independent downstream cases must demonstrate that the proposed records carry enough information without importing inference policy into Jaxstro.

Evidence must also contain adversarial examples where equal covariances correspond to different supports or tail probabilities. Passing a self-consistency check is not evidence that an input probability model describes nature.

Claim boundary¶

This page supplies vocabulary and moment definitions. It does not establish a runtime module, certify a probability model, infer a posterior, combine error sources automatically, or turn numerical error into probability. The proposed owner is future jaxstro.uncertainty, limited to propagation contracts. Scientific meaning remains with the model, data, and downstream workflow that created the input representation.

Connected representations, foundations, and methods¶

Return to Scientific representations to compare uncertainty with units, coordinates, spectra, and parameter-state representations.
Review Probability and distributions for probability mass, density, expectation, covariance, and support.
Use Models, inference, and information to distinguish a model, likelihood, posterior, and information claim.
Compare Random computation and Sampling and resampling for explicit key and sampling mechanics.