Concepts

gpmp contains the numerical GP objects: gpmp.core.Model, covariance functions, parameter-selection functions, diagnosis tools, posterior samplers, and backend operations. gpmp-contrib adds ComputerExperiment, ModelContainer, provided Matérn container classes, sequential strategies, optimization and set-estimation criteria, benchmark problems, and reGP utilities.

Object map

ComputerExperiment

Describes an input domain and one or more callable outputs. Outputs can be objectives, constraints, or generic scalar functions. Constraint outputs may carry bounds that define feasibility.

ModelContainer

Stores one independent gpmp.core.Model per output. Each output entry stores the underlying model, selection criterion, raw parameter vectors, Param object, optimizer report, and resolved prior object when the model class uses REMAP priors.

Provided Matérn container classes

Configure Matérn GP models on top of ModelContainer. They specify the mean form, covariance function, initial-guess procedure, selection criterion, and Param construction.

SequentialPrediction

Stores observations and a model container. It updates model parameters, computes predictions, and draws conditional simulations from the current model.

Sequential strategy classes

Add a candidate set or particle set and a rule for selecting the next evaluation. The implemented rules cover expected improvement, excursion sets, set inversion, and particle-based searches.

Common notation

The guide uses xi for observation points and zi for observed values. Mathematically, write

\[X = (x_1,\ldots,x_n)^\top,\qquad z = (z_1,\ldots,z_n)^\top.\]

For a scalar output, the provided Matérn classes build a Gaussian process

\[Z(x) = m(x) + Z_0(x),\]

where \(Z_0\) is centered and has covariance \(K_\theta(x,x')\). A multi-output ModelContainer stores one independent scalar model per output. Cross-output covariance is not modeled by the provided container classes.

After conditioning on observations, predictions at a point \(x\) are Gaussian:

\[Z(x)\mid (X,z),\theta \sim \mathcal{N}\left(\mu_\theta(x), s_\theta^2(x)\right).\]

The arrays returned by predict are these posterior means and variances, stacked over prediction points and outputs.

Relation to gpmp

The GP model is the standard kriging model used in spatial statistics and computer experiments. Background on covariance modeling and kriging is given by [2, 7]. In gpmp-contrib, the model formulas are not reimplemented. Prediction, likelihood evaluation, REML, REMAP, and posterior parameter sampling are delegated to gpmp.

gpmp-contrib decides how the model is assembled and how state is stored: which covariance class is used, which parameter-selection criterion is called, how optimizer starts are chosen, and how user-readable parameter objects are created.

When to use gpmp-contrib

Use gpmp-contrib when the task involves a computer experiment, a multi-output model container, a sequential design, a provided benchmark problem, or the reGP procedure. Use gpmp directly when implementing a new covariance function, selection criterion, MCMC kernel, backend operation, or diagnostic.