## TR#531: From Hidden Markov Models to Linear Dynamical Systems

### Thomas P. Minka

(1998; revised 7/18/99)
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 difference is that the HMM uses
a discrete state variable with arbitrary dynamics and arbitrary
measurements while the LDS uses a continuous state variable with
linear-Gaussian dynamics and measurements. We show how the
forward-backward equations for the HMM, specialized to linear-Gaussian
assumptions, lead directly to Kalman filtering and Rauch-Tung-Streibel
smoothing. We also investigate the most general possible modeling
assumptions which lead to efficient recursions in the case of
continuous state variables.

This paper is a companion to "Parameter
estimation for linear dynamical systems" by Z. Ghahramani and
G. Hinton.

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