The variational Kalman smoother
Matthew J. Beal, Zoubin Ghahramani · Cambridge University Engineering Department Publications Database · 2001
In this note we outline the derivation of the variational Kalman smoother, in the context of Bayesian Linear Dynamical Systems. The smoother is an efficient algorithm for the E-step in the Expectation-Maximisation (EM) algorithm for linear-Gaussian state-space models. However, inference approximations are required if we hold distributions over parameters. We derive the E-step updates for the hidden states (the variational smoother), and the M-step updates for the parameter distributions. We show that inference of the hidden state is tractable for any distribution over parameters, provided the expectations of certain quantities are available, analytically or otherwise. 1 1 Introduction to variational Linear Dynamical Systems The reader is referred to [1] and [2] for the theoretical framework and motivation for variational Bayesian learning. The joint probability for the state of the hidden, x1:T, and observed, y1:T, variables for a Markov process is given by T∏