No abstract

We consider the problem of computing a Gaussian approximation to the posterior distribution of a parameter given N observations and a Gaussian prior. Owing to the need of processing large sample sizes N, a variety of approximate tractable methods revolving around online learning have flourished over the past decades. In the present work, we propose to use variational inference (VI) to compute a Gaussian approximation to the posterior through a single pass over the data. Our algorithm is a recursive version of variational Gaussian approximation we have called recursive variational Gaussian approximation (R-VGA). We start from the prior, and for each observation we compute the nearest Gaussian approximation in the sense of Kullback-Leibler divergence to the posterior given this observation. In turn, this approximation is considered as the new prior when incorporating the next observation. This recursive version based on a sequence of optimal Gaussian approximations leads to a novel implicit update scheme which resembles the online Newton algorithm, and which is shown to boil down to the Kalman filter for Bayesian linear regression. In the context of Bayesian logistic regression the implicit scheme may be solved, and the algorithm is shown to perform better than the extended Kalman filter, while being less computationally demanding than its sampling counterparts.

We consider the filtering problem of estimating the state of a continuous-time dynamical process governed by a nonlinear stochastic differential equation and observed through discrete-time measurements. As the Bayesian posterior density is difficult to compute, we use variational inference (VI) to approximate it. This is achieved by seeking the closest Gaussian density to the posterior, in the sense of the Kullback-Leibler divergence (KL). The obtained algorithm, called the continuous-discrete variational Kalman filter (CD-VKF), provides implicit formulas that solve the considered problem in closed form. Our framework avoids local linearization, and the estimation error is globally controlled at each step. We first clarify the connections between well known nonlinear Kalman filters and VI, then develop closed form approximate formulas for the CD-VKF. Our algorithm achieves state-of-the-art performances on the problem of reentry tracking of a space capsule.

We consider the problem of computing a Gaussian approximation to the posterior distribution of a parameter given a large number N of observations and a Gaussian prior, when the dimension of the parameter d is also large. To address this problem we build on a recently introduced recursive algorithm for variational Gaussian approximation of the posterior, called recursive variational Gaussian approximation (RVGA), which is a single pass algorithm, free of parameter tuning. In this paper, we consider the case where the parameter dimension d is high, and we propose a novel version of RVGA that scales linearly in the dimension d (as well as in the number of observations N ), and which only requires linear storage capacity in d. This is afforded by the use of a novel recursive expectation maximization (EM) algorithm applied for factor analysis introduced herein, to approximate at each step the covariance matrix of the Gaussian distribution conveying the uncertainty in the parameter. The approach is successfully illustrated on the problems of high dimensional least-squares and logistic regression, and generalized to a large class of nonlinear models.