From Hidden Markov Models to Linear Dynamical Systems
Tom Minka · 1999
Hidden Markov Models (HMMs) and Linear Dynamical Systems (LDSs) are based on the same assumption: a hidden state variable, of which we can make noisy measurements, evolves with Markovian dynamics. Both have the same independence diagram and consequently the learning and inference algorithms for both have the same structure. The only dierence is that the HMM uses a discrete state variable with arbitrary dynamics and arbitrary measurements while the LDS uses a continuous state variable with linearGaussian dynamics and measurements. We show how the forward-backward equations for the HMM, specialized to linear-Gaussian assumptions, lead directly to Kalman ltering and Rauch-Tung-Streibel smoothing. We also investigate the most general possible modeling assumptions which lead to ecient recursions in the case of continuous state variables. 1 Introduction Both HMMs and LDSs specify a joint probability distribution over states s 1 ::s T and measurements x 1 ::x T . In both cases, the distri...