Ensemble propagation

Use this page when the distribution-generation process is available as explicit draws and nonlinear or multimodal output structure matters beyond a local moment approximation.

The scientific question¶

What output variation results when an explicitly generated collection of inputs is mapped through the same scientific function? Ensemble propagation evaluates $f(\mathbf{x}_n)$ for each member and summarizes or retains the resulting empirical distribution. It can expose skewness, tail behavior, or separated modes that a single covariance misses.

An ensemble only represents the distribution-generation process used to create it. Independent prior draws, posterior-chain states, bootstrap replicates, randomized quadrature points, and perturbed numerical resolutions have different meanings even when they have the same array shape. The generator, conditioning information, weights, dependence, and random-key lineage must accompany the values.

Mathematical objects¶

Let $\mathbf{x}_n\in\mathbb{R}^{d}$ for $n=1,\ldots,N$ be explicit input members, and let $\mathbf{y}_n=f(\mathbf{x}_n)\in\mathbb{R}^{m}$ . For the ordinary unbiased sample covariance below, require N >= 2, equal weights, and members that justify the usual independent-sample interpretation. Define $\boldsymbol{\epsilon}_n=\mathbf{y}_n-\widehat{\boldsymbol{\mu}}_y$ .

Array axes are semantic. One axis indexes ensemble members; remaining axes describe one scientific output. Units belong to output components, and covariance entry $(a,b)$ has units $[Y_a][Y_b]$ . A batch axis used for device execution must not be confused with an independent replicate axis.

Weighted ensembles require normalized weights and a covariance denominator matched to their sampling design. Correlated Markov-chain states require autocorrelation or effective-sample-size reasoning. Neither case is described by blindly replacing $N$ in the ordinary formula.

Core derivation¶

For equal-weight members, the empirical mean and Bessel-corrected sample covariance are

\widehat{\boldsymbol{\mu}}_y=\frac{1}{N}\sum_{n=1}^{N}f(\mathbf{x}_n), \qquad \widehat{\mathbf{C}}_y=\frac{1}{N-1}\sum_{n=1}^{N} \boldsymbol{\epsilon}_n\boldsymbol{\epsilon}_n^{\mathsf{T}}, \qquad N\geq 2.

(1)

Equation (1) uses $N-1$ because estimating the sample mean imposes one linear constraint on the centered residuals. For independent draws with finite second moments, this makes the sample covariance unbiased for the population covariance. It does not make the sample standard deviation, inverse covariance, or nonlinear functions of the covariance unbiased.

The empirical mean has Monte Carlo error that commonly scales as $N^{-1/2}$ under independent finite-variance sampling, but the coefficient and applicability depend on the observable and generator. Tail probabilities, high-dimensional covariance, and rare-event observables may converge much more slowly in practice.

Failure modes and interpretation limits¶

Reusing one PRNG key can duplicate or correlate members while preserving plausible shapes and finite values.
Treating correlated chain states as independent understates estimator uncertainty.
Ignoring nonuniform importance weights changes the represented distribution.
Small ensembles produce noisy, rank-deficient covariance estimates; rank is at most $N-1$ after centering.
Rare modes or tails absent from the generator cannot appear after propagation.
Device batching can exceed memory because both inputs and outputs may be retained; streaming moments reduce memory but discard sample-level diagnostics.
Adaptive stopping based on observed ensemble values changes the sampling design and must be recorded.
Numerical failures in selected members cannot be silently removed without changing weights and support.

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 surface may define member-axis metadata, deterministic key-splitting policy, fixed-shape mapped evaluation, chunked accumulation, failure masks, and provenance for the generator and executed map. It must not implement posterior sampling, hide keys, infer independence from array shape, or relabel one ensemble as a universally valid uncertainty distribution.

Evidence required before implementation¶

Tests must reproduce analytic transformed moments for simple laws and maps, verify the $N-1$ covariance convention on hand-calculated examples, and demonstrate Monte Carlo convergence across independent replications rather than one lucky seed. Deterministic replay must cover key roots, split order, member ordering, device batching, and chunk sizes. Weighted and dependent inputs must either use explicitly validated formulas or fail closed as unsupported.

Performance evidence must separate compilation, evaluation, transfer, and retained memory. Scientific evidence must report ensemble size, effective rank, failed-member policy, generator identity, component units, and convergence of the actual target observable. Agreement with itself after reshaping is not an independent validation.

Claim boundary¶

An ensemble supports claims about its documented generation process and finite sample, not every distribution consistent with its members. This guide does not provide a sampler, posterior, likelihood, convergence diagnosis for arbitrary chains, or scientific acceptance policy. A future Jaxstro owner could propagate and audit explicit ensembles only; inference and domain interpretation remain with their established owners.

Connected representations, foundations, and methods¶

Begin with What uncertainty represents to classify the ensemble’s meaning.
Return to Scientific representations for shape, unit, and provenance context.
Review Probability and distributions for expectation and sample statistics.
Use Random computation, Sampling and resampling, and Linearized propagation to compare key mechanics, sampling mechanics, and local propagation.