The Auxiliary Variable Trick for deriving Kalman Smoothers

We present a Forward-Backward Kalman smoother derivation that functions for small observation noise. Whilst this smoother can be found by judicious transformation of the standard Forward-Backward equations, we introduce an auxiliary variable trick which greatly simplifies the derivation, based on the probabilistic interpretation of the Kalman Filter, allowing one to work directly with moments of the distribution. The trick is of potential interest for the simple derivation of smoothing type inference in other related systems.

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