Kernkonzepte
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.
Zusammenfassung
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.