  The Kalman filter (KF) is one of the most widely used tools for data
assimilation and sequential estimation. In this work, we show that the state
estimates from the KF in a standard linear dynamical system setting are
equivalent to those given by the KF in a transformed system, with infinite
process noise (i.e., a "flat prior") and an augmented measurement space. This
reformulation -- which we refer to as augmented measurement sensor fusion (SF)
-- is conceptually interesting, because the transformed system here is
seemingly static (as there is effectively no process model), but we can still
capture the state dynamics inherent to the KF by folding the process model into
the measurement space. Further, this reformulation of the KF turns out to be
useful in settings in which past states are observed eventually (at some lag).
Here, when the measurement noise covariance is estimated by the empirical
covariance, we show that the state predictions from SF are equivalent to those
from a regression of past states on past measurements, subject to particular
linear constraints (reflecting the relationships encoded in the measurement
map). This allows us to port standard ideas (say, regularization methods) in
regression over to dynamical systems. For example, we can posit multiple
candidate process models, fold all of them into the measurement model,
transform to the regression perspective, and apply $\ell_1$ penalization to
perform process model selection. We give various empirical demonstrations, and
focus on an application to nowcasting the weekly incidence of influenza in the
US.
