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Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces

In a variety of applications it is important to extract information from a probability measure $μ$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $ν$, from within a simple class of measures, which approximates $μ$. This problem is studied in the case where the Kullback-Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted and the particular case where $ν$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimising sequences. Furthermore, parameterisation of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularisation is explained, and a regularised minimisation is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms.