  In a variety of applications it is important to extract information from a
probability measure $\mu$ 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 $\nu$, from
within a simple class of measures, which approximates $\mu$. 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 $\nu$ 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.
