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Taylor approximation for uncertain-input Gaussian-process prediction

Algorithms AIMech:GaussianProcessTaylorUncertainInputPrediction DRAFT

Aliases: Taylor Gaussian-process uncertainty propagation

A local Gaussian-process prediction algorithm that expands a fixed posterior mean to first order and its latent variance to second order about the mean of a prescribed Gaussian test input, integrates those expansions, and uses the resulting approximate moments for a Gaussian predictive representation and iterative covariance propagation.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and scoped synonyms and 541 native IDs, names and aliases against the proposed ID, name, descriptive alias and Taylor uncertain-input GP terms: no normalized identity match. Ignored- and hidden-inclusive full-text searches in the recovery, older discovery and original Documents checkouts returned 98, 0 and 0 lines. All 56 distinct matched strings were read, including the full generated search-index match; they describe the pending Taylor lead or the already represented exact Gaussian-input integration procedure. Read 37 semantic AIO rows, 77 native name/definition neighbors, six full annotated CLASS nodes with incident edges/generated records and five complete contextual native records. The scoped method approximates posterior mean and variance functions by local Taylor expansions before integrating over a Gaussian test input. It differs from exact squared-exponential input integration, generic GP regression, derivative observations used as training data and correlation-aware trajectory linearization. Absence is bounded to the inspected pin and maintained corpus; the alias does not claim all GP uncertainty propagation methods are identical.

Claims & Grounded Evidence

5 Claims

Mechanism #local-moments

For a fixed trained GP posterior with differentiable scalar mean m(x) and twice-differentiable latent variance v(x), take a prescribed Gaussian test input X~N(mu,Sigma) independent of the posterior function conditional on the training data and fixed hyperparameters. The source expands m to first order and v to second order about mu. Integrating these truncated expressions gives the predictive mean approximation m(mu) and latent variance approximation v(mu)+0.5*tr(H_v(mu)*Sigma)+grad(m)(mu)^T*Sigma*grad(m)(mu). This combines the expected conditional variance and the variance of the linearized conditional mean. A Gaussian with these moments is an additional approximation to the predictive law, provided the resulting variance is valid.

supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Sections 3.2–3.4, Equations 12–22, PDF pp. 8–9
supports Source: Gaussian Process Priors with Uncertain Inputs Application to Multiple-Step Ahead Time Series Forecasting · Locator: Section 3.1, Equations 6–9, PDF p. 3

Application #lag-covariance

For iterative autoregressive prediction the report carries a full Gaussian covariance of the lagged outputs. The local expansion gives Cov(X,f(X)) approximately Sigma*grad(m)(mu), with the transpose used for Cov(f(X),X). This cross-covariance forms the new row and column while the retained lag block shifts forward. If predicting a future noisy output rather than the latent function value, add its independent observation-noise variance once to the new output diagonal. Training noise already used in the GP posterior and this future output noise play different roles. Using only independent lag variances is a different approximation.

supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Sections 4.1–4.2, PDF pp. 11–13; Equations 32–35 and the lag covariance matrix, PDF p. 13

Limitation #mean-curvature

A second-order Taylor expansion of the conditional variance is not a general assertion of second-order accuracy for the predictive moments: the source retains only a first-order expansion of the conditional mean. A curator unit-kernel example conditioned on the exact observation f(0)=1 and X~N(0,s) gives Taylor mean 1 and Taylor variance s, whereas exact integration gives mean 1/sqrt(1+s) and variance s/(1+s). The mean error is of order s as s approaches zero, even though the posterior-mean gradient vanishes at the expansion point. At Sigma=0 the formulas reduce to ordinary fixed-input GP prediction.

supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Sections 3.3–3.4, Equations 14–22, PDF pp. 8–9
supports Source: Gaussian Process Priors with Uncertain Inputs Application to Multiple-Step Ahead Time Series Forecasting · Locator: Section 3.1, exact squared-exponential integrals, Equations 10–13, PDF p. 4

Limitation #variance-validity

The local Taylor variance is not guaranteed nonnegative for arbitrary input uncertainty. A curator calculation for a unit-amplitude, unit-lengthscale squared-exponential GP conditioned on an exact f(0)=0, with input mean 1 and variance 2, gives Taylor variance 1-3/e<0 while the exact integrated variance is 1-exp(-1/5)/sqrt(5)>0. This deliberately nonlocal case limits an unrestricted covariance claim; it is not an empirical reproduction or a refutation of a justified small-uncertainty approximation. An invalid computed covariance cannot be interpreted as a Gaussian predictive distribution without a separately justified correction.

supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Section 3.4, Equations 18–22, PDF p. 9
supports Source: Gaussian Process Priors with Uncertain Inputs Application to Multiple-Step Ahead Time Series Forecasting · Locator: Section 3.1, Equations 8–13, PDF pp. 3–4

Scope #distinct-exact-and-correlated-methods

The local Taylor procedure is distinct from analytic squared-exponential Gaussian-input moment integration, which the technical report lists as future work and the proceedings paper presents separately. Moreover, propagating a Gaussian lag covariance while repeatedly treating the current input as independent of the uncertain function does not preserve all dependence created by one shared dynamics function across a trajectory. The later correlation-aware trajectory approaches address that different approximation; this record does not identify them with the earlier local Taylor method.

supports Source: Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction · Locator: Conclusion and future work, PDF p. 17
supports Source: Gaussian Process Priors with Uncertain Inputs Application to Multiple-Step Ahead Time Series Forecasting · Locator: Section 3.1, Taylor Equations 8–9 versus exact Equations 10–13, PDF pp. 3–4
supports Source: On Simulation and Trajectory Prediction with Gaussian Process Dynamics · Locator: Introduction and Sections 2–3, PDF pp. 1–4; proposed trajectory methods, PDF pp. 5–7

⚠️ Curated Limitations & Negative Results

The method is a local approximation requiring sufficient smoothness and input uncertainty small enough for the retained expansions to be useful. Gaussian test-input independence, fixed training locations/data and fixed hyperparameters are explicit assumptions; uncertain training locations and hyperparameter integration are not solved here.
The first-order mean omits the curvature term 0.5*tr(H_m*Sigma). The scalar negative-variance example also prevents a universal positive-semidefinite covariance claim. No general error bound, calibration guarantee or automatic covariance-repair procedure is asserted.
The technical report changes its squared-exponential width notation: early widths are lengthscales in a denominator, while Equation 25 uses inverse squared lengthscales. Any implementation must choose one consistent parameterization. The record uses posterior derivatives directly and does not copy incompatible width symbols across sections.
Full lag covariance handles correlation between uncertain regressor components, but does not retain the complete shared-function trajectory dependence. Gaussian closure also omits higher moments. A local covariance formula is not exact nonlinear trajectory propagation.
The source forecasting mean can coincide with a mean-only rollout because this particular mean approximation is m(mu); the covariance correction changes the uncertainty estimate. Source comparisons on synthetic dynamics and pH data do not establish universal accuracy, superiority or calibrated coverage, and their numerical results are not curated here.
The exact-observation unit-kernel examples are curator analytic checks, not fitted-model or benchmark reproductions. No author implementation, training, paid provider or stochastic simulation was run. The record remains DRAFT after author self-review.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:RegressionAnalysis` Regression Analysis	related_match	The algorithm approximates a fitted GP response prediction under uncertain test inputs. RegressionAnalysis denotes the task context, not this local integration procedure or an equivalent algorithm.
`AIO:TimeSeriesForecasting` Time Series Forecasting	related_match	The primary report explicitly develops the Taylor construction for iterative multiple-step time-series prediction with lag covariance. This is an application relationship, not a claim that the forecasting task is a superclass-equivalent algorithm.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`girard-tr2002`	Gaussian Process priors with Uncertain Inputs: Multiple-Step-Ahead Prediction Agathe Girard, Carl Edward Rasmussen, Roderick Murray-Smith	author technical report full_text	ID: TR-2002-119 sha256:f2e78afd674d…
`girard2002`	Gaussian Process Priors with Uncertain Inputs Application to Multiple-Step Ahead Time Series Forecasting Agathe Girard, Carl Edward Rasmussen, Joaquin Quiñonero Candela, Roderick Murray-Smith	primary paper full_text	ID: https://papers.nips.cc/paper_files/paper/2002/hash/f3ac63c91272f19ce97c7397825cc15f-Abstract.html sha256:b8b19f828d8d…
`hewing2020`	On Simulation and Trajectory Prediction with Gaussian Process Dynamics Lukas Hewing, Elena Arcari, Lukas P. Fröhlich, Melanie N. Zeilinger	primary paper full_text	ID: https://proceedings.mlr.press/v120/hewing20a.html sha256:e03bc1f12364…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T21:22:08Z

Updated: 2026-10-12T21:22:08Z

Source File: data/curated/GaussianProcessTaylorUncertainInputPrediction.yaml