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Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

Nadège Polette, Olivier Le Maître, Pierre Sochala, Alexandrine Gesret

Abstract

This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{è}ve(KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only onthe field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusionproblem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry.

Change of Measure for Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

Abstract

This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Lo{è}ve(KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only onthe field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusionproblem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry.
Paper Structure (27 sections, 65 equations, 18 figures, 1 table)

This paper contains 27 sections, 65 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bayesian field inference
  3. Bayes' formulation
  4. Dimension reduction of random fields
  5. Change of measure method
  6. Reference basis
  7. Change of coordinates
  8. Change of measure
  9. for the change of measure
  10. Accelerating sampling in the framework
  11. Polynomial chaos surrogates
  12. Surrogate models for likelihood computation
  13. Surrogate models for prior computation
  14. Application to a transient diffusion problem
  15. Case presentation
  16. ...and 12 more sections

Figures (18)

  • Figure 1: Workflow of the (left) and the (right) methods. The quantities that are specific to the method are indicated in red.
  • Figure 2: td case - Eigenvalues decay for different correlation lengths $l$ as indicated.
  • Figure 3: td case - Diagonal terms of $\Sigma(l)$ corresponding to the prior variance of $\boldsymbol{\xi}$.
  • Figure 4: td case - Prior Covariance $\Sigma(l)$ for two correlation lengths. Projection on the $6th$ and $8th$ coordinates.
  • Figure 5: td case - values (left) according order ($r=8$), (right) according number of terms ( order $=15$). $1,000$ random validation points.
  • ...and 13 more figures