Solving Bayesian inference problems approximately with variational approaches can provide fast and accurate results. Capturing correlation within the approximation requires an explicit parametrization. This intrinsically limits this approach to either moderately dimensional problems, or requiring the strongly simplifying mean-field approach. We propose Metric Gaussian Variational Inference (MGVI) as a method that goes beyond mean-field. Here correlations between all model parameters are taken into account, while still scaling linearly in computational time and memory. With this method we achieve higher accuracy and in many cases a significant speedup compared to traditional methods. MGVI is an iterative method that performs a series of Gaussian approximations to the posterior. We alternate between approximating the covariance with the inverse Fisher information metric evaluated at an intermediate mean estimate and optimizing the KL-divergence for the given covariance with respect to the mean. This procedure is iterated until the uncertainty estimate is self-consistent with the mean parameter. We achieve linear scaling by avoiding to store the covariance explicitly at any time. Instead we draw samples from the approximating distribution relying on an implicit representation and numerical schemes to approximately solve linear equations. Those samples are used to approximate the KL-divergence and its gradient. The usage of natural gradient descent allows for rapid convergence. Formulating the Bayesian model in standardized coordinates makes MGVI applicable to any inference problem with continuous parameters. We demonstrate the high accuracy of MGVI by comparing it to HMC and its fast convergence relative to other established methods in several examples. We investigate real-data applications, as well as synthetic examples of varying size and complexity and up to a million model parameters.
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