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Monotonic Gaussian process with virtual derivatives

Algorithms AIMech:MonotonicGaussianProcessWithVirtualDerivatives DRAFT

Aliases: GP with monotonicity information, Gaussian process with monotonicity information

A Gaussian-process inference procedure that promotes monotonicity in selected input coordinates by combining observed outputs with finite virtual derivative-sign factors and approximating the resulting posterior using expectation propagation.

Identity Notes & Concept Scope Compared all 442 pinned AIO IDs, labels and scoped synonyms and 538 native IDs, names and aliases against the proposed ID, names and descriptive variants: no normalized identity match. Full ignored- and hidden-inclusive searches of the recovery worktree, older discovery worktree and restored original Documents checkout returned 392, 362 and 361 lines. Candidate-specific excerpts and matched paths were inspected: the virtual-derivative references are contextual mentions in the Hammerstein–Wiener record and its pending predictor cohort; most broader monotonicity hits concern optimization objectives. Read all 35 semantic AIO rows, 88 native name/definition neighbors, five complete annotated AIO CLASS nodes with incident edges, their generated records and three native neighbors. General Gaussian process regression does not specify derivative-sign factors or EP sites. FABOLAS uses a size-dependent GP surrogate for resource selection; the Hammerstein–Wiener record defines a cascade family. AIO regression and binary classification are tasks; GLM and graphical-model definitions do not supply this inference procedure. This absence assessment is bounded to the inspected snapshot and records.

Claims & Grounded Evidence

4 Claims

Mechanism #joint-values-and-derivatives

A differentiable GP prior supplies joint covariance blocks for function values and selected partial derivatives by differentiating its kernel. The inspected formulation uses a squared-exponential kernel with coordinate-specific lengthscales. These latent derivatives are coupled to the observed function values; their virtual sign information is distinct from measuring a derivative magnitude. For a unit one-dimensional squared-exponential kernel, the derivative–derivative covariance at separation d is (1-d²)*exp(-d²/2), which can be zero or negative away from the same point.

supports Source: Gaussian processes with monotonicity information · Locator: Section 2, printed p. 646 / PDF p. 2; Section 3 Equation 6, p. 647 / PDF p. 3; kernel derivative check in research/discovery/monotonic-gp-boundaries.py

Mechanism #probit-sites-and-ep

At each chosen location and input coordinate, positive derivative information contributes Phi(f_prime/nu), with nu>0 and Phi the standard-normal CDF. EP replaces these non-Gaussian factors with Gaussian sites and approximates the joint posterior. Gaussian observation likelihoods are used for regression; binary probit classification applies EP to class-observation factors as well. Hyperparameters maximize an approximate joint marginal likelihood in the inspected procedure rather than being integrated out.

supports Source: Gaussian processes with monotonicity information · Locator: Section 3, Equations 5–10 and subsequent inference/prediction equations, printed pp. 647–648 / PDF pp. 3–4

Mechanism #virtual-point-placement

The source proposes grids in low dimensions, selected observed covariate locations in higher dimensions, and iterative placement where the current predictive derivative has high probability of being negative. Points can be added, moved or removed after conditioning. Dense computation scales cubically in the combined number of real and virtual observations, and a grid grows exponentially with input dimension; spacing relative to the fitted lengthscale affects how constraints influence intervening locations.

supports Source: Gaussian processes with monotonicity information · Locator: Section 3.1, printed pp. 648–649 / PDF pp. 4–5; Section 5, p. 652 / PDF p. 8

Limitation #approximate-monotonicity

Finite derivative sites and an EP approximation do not guarantee a globally monotone function. As a logical finite-site counterexample, f(x)=2x³-3x²+x has derivative 1 at x=0 and x=1, but -1/2 at x=1/2. Finite nu also gives positive likelihood to negative slopes. Even the Gaussian matching a standard normal truncated to a positive derivative retains about 0.0928 probability below zero. This last value is a curator one-site limiting calculation, not a measured violation rate for the paper’s models or a run of its EP implementation.

supports Source: Gaussian processes with monotonicity information · Locator: Section 3 Equation 5 and Gaussian approximation, printed p. 647 / PDF p. 3; Section 4.2 and conclusion, pp. 651–652 / PDF pp. 7–8; curator calculations in research/discovery/monotonic-gp-boundaries.py

⚠️ Curated Limitations & Negative Results

This record denotes the Riihimäki–Vehtari virtual-derivative/probit/EP inference procedure for regression and binary classification. It is not every shape-constrained GP, an isotonic estimator, a construction-enforced monotone random process or a generic EP record. The inspected paper uses positive derivative information in selected coordinates; any decreasing-coordinate extension must state its sign convention.
The derivative covariance requires a sufficiently differentiable kernel. Finite site placement, approximate Gaussian inference and point-estimated hyperparameters do not certify monotonicity between sites, calibrated uncertainty, convergence to a global optimum or correct prior knowledge.
The source reports lower errors for some smooth synthetic functions and worse errors for its step-function examples, attributing difficulties to smoothness and Gaussian approximation near truncated derivative boundaries. Its health-care classification example is an application with an assumed age trend, not evidence of causal effects or general clinical benefit.
The cubic counterexample checks the logic of finite derivative constraints, not a sampled squared-exponential GP trajectory. The half-normal calculation is a one-site hard-sign limit and is not the finite-nu multi-site empirical violation probability. No source implementation or empirical experiment was executed.
All primary PDF pages were read. The author GPstuff demonstration is a discovered implementation lead; its complete implementation and runtime behavior were not audited. This record remains DRAFT and author self-review is not independent scientific approval.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:RegressionAnalysis` Regression Analysis	related_match	The Gaussian-observation variant estimates a response relationship while adding derivative-sign information. The target names a broad task, whereas this record specifies an inference procedure; the mapping does not assert identity or an upstream subclass.
`AIO:BinaryClassification` Binary Classification	related_match	The source extends the procedure to two-class probit observations, approximating both class and derivative factors. This is a method/task relationship, not exact identity or a claim about every classification method.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`riihimaki-vehtari2010`	Gaussian processes with monotonicity information Jaakko Riihimäki, Aki Vehtari	primary paper full_text	ID: https://proceedings.mlr.press/v9/riihimaki10a.html sha256:0221482450da…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T15:00:52Z

Updated: 2026-10-12T15:00:52Z

Source File: data/curated/MonotonicGaussianProcessWithVirtualDerivatives.yaml