insight - Computational Complexity - # Bayesian Field Inversion with Hierarchical Hyperparameters Sampling

Core Concepts

The core message of this paper is to introduce a novel change of measure (CoM) approach to efficiently sample the joint posterior distribution of a scalar field and its hyperparameters in a Bayesian inference framework.

Abstract

The 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 due to the infinite dimensionality of the field, which requires a finite parametrization involving hyperparameters.

The authors introduce a change of measure (CoM) approach to overcome the limitations of the previously proposed change of coordinates (CoC) method. In the CoM approach, the hyperparameter dependencies are transferred to the prior distribution of the field coordinates in a fixed reference basis, rather than modifying the field representation itself. This leads to a hierarchical Bayesian formulation where the likelihood only depends on the field coordinates, while the prior distribution depends on the hyperparameters.

The CoM approach avoids the need to recompute the eigenmodes of the autocovariance function at each MCMC step, which was a major computational bottleneck in the CoC method. The authors also leverage polynomial chaos expansions to construct surrogate models for the forward model predictions and the CoM-derived quantities, further accelerating the MCMC sampling.

The performance of the CoM method is assessed on two test cases: a transient diffusion problem and a seismic traveltime tomography problem. The results show that the CoM approach is consistent with the CoC method, while being more efficient and providing a better physical interpretation of the role of the hyperparameters.

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Stats

The noise level σε is the standard deviation of the observation noise ε = dobs - M(ftrue), where dobs are the observations and M(ftrue) are the true model predictions.
The true velocity field vtrue is adapted from [53] and depends only on the vertical coordinate.

Quotes

"The aim of this paper is to introduce a new method to alleviate the difficulties of the CoC method by transferring the q-dependency to the prior distribution of the field coordinates."
"The advantage of our formulation is that the posterior distribution only depends on q through the prior distribution."

Deeper Inquiries

The CoM method is specifically designed for problems where the field and hyperparameters follow Gaussian distributions. When dealing with non-Gaussian random fields or non-Gaussian observation noise models, the performance of the CoM method may be affected.
For non-Gaussian random fields, the assumption of Gaussianity in the CoM method may lead to inaccuracies in representing the field distribution. Non-Gaussian fields may have heavier tails or skewness that cannot be captured effectively by a Gaussian distribution. This could result in biased estimates and unreliable inference results.
Similarly, non-Gaussian observation noise models can introduce complexities in the likelihood function. If the noise is not normally distributed, the likelihood calculation in the CoM method may not accurately capture the true data generation process. This can lead to incorrect posterior distributions and inference outcomes.
In such cases, modifications to the CoM method would be necessary to accommodate non-Gaussianity in both the field and observation noise models. This could involve using alternative probabilistic models or sampling techniques that are more suitable for non-Gaussian data.

The CoM approach has limitations related to the dimensionality of the field and the number of hyperparameters.
In terms of field dimensionality, as the number of dimensions increases, the computational complexity of the CoM method also grows. The calculation of the covariance matrix and the sampling of the joint posterior distribution become more challenging in high-dimensional spaces. This can lead to increased computational costs and difficulties in achieving convergence in the MCMC sampling.
Regarding the number of hyperparameters, the CoM method may face limitations when dealing with a large number of hyperparameters. As the hyperparameter space expands, the prior distribution of the field coordinates becomes more sensitive to variations in the hyperparameters. This can result in a more complex and intricate prior distribution, making it harder to sample from and leading to potential issues with the inference process.
Therefore, the CoM approach may encounter practical constraints when applied to problems with high-dimensional fields or a large number of hyperparameters. Careful consideration of these limitations is essential when implementing the CoM method in such scenarios.

The CoM framework, as presented in the context, follows a hierarchical structure where the hyperparameters influence the prior distribution of the field coordinates. While this hierarchical approach is effective for many problems, there are possibilities to extend the CoM method to handle more complex dependencies between the field and hyperparameters.
One potential extension could involve incorporating non-linear relationships between the field and hyperparameters. By introducing non-linear transformations or interactions in the prior distribution, the CoM framework could capture more intricate dependencies that go beyond the simple hierarchical structure. This would require adapting the sampling algorithms and the surrogate models to accommodate the increased complexity.
Additionally, the CoM method could be extended to handle dynamic or time-varying dependencies between the field and hyperparameters. By incorporating temporal dependencies or evolving relationships, the CoM framework could be applied to problems where the field and hyperparameters change over time. This extension would involve developing specialized sampling strategies and surrogate models to account for the dynamic nature of the dependencies.
Overall, the CoM framework has the potential for further development to address more complex and nuanced relationships between the field and hyperparameters, beyond the hierarchical structure considered in the current work.

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