Concave Gaussian Variational Approximations for Inference in Large-Scale Bayesian Linear Models
Edward Challis, David Barber · 2011
Two popular approaches to forming bounds in approximate Bayesian inference are local variational methods and minimal Kullback-Leibler divergence methods. For a large class of models we explicitly relate the two approaches, showing that the local variational method is equivalent to a weakened form of Kullback-Leibler Gaussian approximation. This gives a strong motivation to develop efficient methods for KL minimisation. An important and previously unproven property of the KL variational Gaussian bound is that it is a concave function in the parameters of the Gaussian for log concave sites. This observation, along with compact concave parametrisations of the covariance, enables us to develop fast scalable optimisation procedures to obtain lower bounds on the marginal likelihood in large scale Bayesian linear models. 1 BAYESIAN MODELS For parameter w and data D, a large class of Bayesian models describe posteriors of the form p(w|D) = 1 N (w µ, Σ) φ(w), (1.1) Z∫ Z = N (w µ, Σ) φ(w)dw for a Gaussian factor N (w µ, Σ) and positive potential function φ(w). This class includes generalised linear models, see e.g. Hardin and Hilbe (2007), and Gaussian noise models in inverse modeling, see e.g. Wipf and Nagarajan (2009). A classic example is Bayesian logistic regression in which N (w µ, Σ) is the