  Latent Gaussian models (LGMs) are widely used in statistics and machine
learning. Bayesian inference in non-conjugate LGMs is difficult due to
intractable integrals involving the Gaussian prior and non-conjugate
likelihoods. Algorithms based on variational Gaussian (VG) approximations are
widely employed since they strike a favorable balance between accuracy,
generality, speed, and ease of use. However, the structure of the optimization
problems associated with these approximations remains poorly understood, and
standard solvers take too long to converge. We derive a novel dual variational
inference approach that exploits the convexity property of the VG
approximations. We obtain an algorithm that solves a convex optimization
problem, reduces the number of variational parameters, and converges much
faster than previous methods. Using real-world data, we demonstrate these
advantages on a variety of LGMs, including Gaussian process classification, and
latent Gaussian Markov random fields.
