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Conditioning of Banach Space Valued Gaussian Random Variables: An Approximation Approach Based on Martingales


Core Concepts
The core message of this article is to provide a general framework for conditioning one Banach space valued Gaussian random variable with respect to another, including an approximation scheme based on martingales that allows for efficient computation of the conditional means and covariances.
Abstract
The article investigates the conditioning of Banach space valued Gaussian random variables. The key insights are: It generalizes the well-known finite dimensional results on conditioning Gaussian random variables to the infinite dimensional Banach space setting. Specifically, it provides formulas for computing the conditional means and covariances. To handle the case of infinite dimensional observations, the article introduces the concept of "filtering sequences" - a sequence of finite dimensional projections that converge to the full observation space. Using this, it shows that the conditional distributions of the finite dimensional approximations converge weakly to the true conditional distribution. The article establishes that the conditional means and covariances can be computed using the same structure as in the finite dimensional case, but with the covariance operators replacing the covariance matrices. It constructs a specific version of the regular conditional probability that has desirable properties, including the conditional mean being a version of the conditional expectation. The results leverage a Banach space valued martingale approach, which allows handling the general Banach space setting, going beyond previous works that were restricted to Hilbert spaces. The article applies the general theory to the specific case of conditioning Gaussian processes, providing uniform convergence results for the conditional means and covariances.
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by Ingo Steinwa... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03453.pdf
Conditioning of Banach Space Valued Gaussian Random Variables

Deeper Inquiries

How can the results of this article be extended to non-Gaussian or non-linear conditioning problems

The results of the article can be extended to non-Gaussian or non-linear conditioning problems by adapting the framework to handle different types of distributions and relationships between random variables. For non-Gaussian conditioning, the key would be to generalize the concept of regular conditional probabilities and conditional expectations to accommodate non-Gaussian distributions. This may involve using different mathematical tools and techniques to characterize the conditional distributions and expectations accurately. For non-linear conditioning problems, the framework would need to incorporate non-linear transformations and relationships between the random variables, which may require more complex mathematical formulations and computations.

What are the potential applications of the developed framework beyond Gaussian processes, e.g. in inverse problems or uncertainty quantification

The developed framework has potential applications beyond Gaussian processes in various fields such as inverse problems and uncertainty quantification. In inverse problems, the framework can be used to model and analyze the conditional distributions of random variables in scenarios where observations are limited or noisy. This can help in solving inverse problems more effectively by providing insights into the uncertainty associated with the solutions. In uncertainty quantification, the framework can aid in quantifying and managing uncertainties in complex systems by providing a systematic approach to conditioning random variables based on observations or partial information.

Can the martingale-based approach be further generalized to handle other types of conditioning beyond the Gaussian setting

The martingale-based approach used in the article can potentially be generalized to handle other types of conditioning beyond the Gaussian setting by adapting the martingale concept to different probability distributions and random variable relationships. By extending the martingale idea to non-Gaussian or non-linear settings, it may be possible to develop a more versatile and robust framework for conditioning random variables in various contexts. This generalization would involve exploring the properties of martingales in different probability spaces and adapting the martingale approximation approach to suit the specific characteristics of the non-Gaussian or non-linear conditioning problems.
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