How deeply does Bayesian Optimization in Action cover multifidelity matérn kernel?

Defines covariance behavior and implementation usage, but defers mathematical details to BoTorch documentation.

Multifidelity matérn kernel

As explained in Bayesian Optimization in Action

A multifidelity Matérn kernel is a Matérn covariance function extended to model observations from multiple fidelity levels and their correlations.

A multifidelity Matérn kernel extends Matérn covariance to observations gathered from several fidelity levels. It uses each point’s fidelity correlation to decide how strongly that observation informs nearby predictions, making inexpensive but inexact evaluations useful when optimizing an expensive black-box objective. section 9.2.2 section 9.7

The kernel supports multifidelity Gaussian-process modeling by:

Representing fidelity correlation as an extra feature column, with values between 0 and 1 indicating similarity to the ground truth. section 9.7
Producing higher covariance, and therefore greater uncertainty reduction, for nearby observations with high correlation. section 9.2.2
Allowing low-correlation observations to leave posterior uncertainty relatively high while still contributing to the model. section 9.2.2
Supporting a specialized BoTorch Gaussian-process class that is trained by maximizing the likelihood of the observations. section 9.2.2

Low- and high-correlation observations

A GP trained only on low-fidelity observations becomes less uncertain when their correlation is 0.9 than when it is 0.5. With both fidelity levels, mean predictions pass through high-fidelity observations while low-fidelity points still reduce uncertainty.

from botorch.models.gp_regression_fidelity
import SingleTaskMultiFidelityGP         #1
 
model = SingleTaskMultiFidelityGP(
train_x_full, train_y, data_fidelity=1)  #2

Instantiate BoTorch’s SingleTaskMultiFidelityGP with the full feature matrix, response values, and the column containing fidelity correlation values. section 9.2.2

How Bayesian Optimization in Action covers it

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Related concepts

multifidelity gaussian processused by · A multifidelity Gaussian process is a Gaussian-process model that represents correlations between observations obtained from different fidelity levels of an objective function.
matérn kernelkind of · A Matérn kernel is a covariance function whose smoothness is controlled by a parameter that determines the finite differentiability of functions drawn from the associated Gaussian process.
multifidelity optimizationused by · Multifidelity optimization combines evaluations of an objective at different levels of accuracy and cost to reduce optimization expense.
fidelity correlationconfigured by · A fidelity correlation is a parameter describing how informative observations at one fidelity are about function values at another fidelity.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.