The Variational Kalman Smoother

In this note we outline the derivation of the variational Kalman smoother, in the context of Bayesian Linear Dynamical Systems. The smoother is an e#cient algorithm for the E-step in the ExpectationMaximisation (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