Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function

TL;DR

This paper introduces a method combining coordinate transformation and Polynomial Chaos expansions to efficiently perform Bayesian inference of Gaussian processes with uncertain covariance hyper-parameters, improving the accuracy of inferred fields.

Contribution

It develops a generalized Karhunen-Loève expansion and employs Polynomial Chaos to incorporate hyper-parameter uncertainty into Bayesian inference, reducing computational complexity.

Findings

Improved inference accuracy when including hyper-parameter uncertainty.

Efficient Bayesian inference using Polynomial Chaos expansions.

Validated on a transient diffusion equation with noisy data.

Abstract

This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-\Loeve expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Lo\`{e}ve coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Lo\`{e}ve expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.