  We consider the Bayesian approach to the linear Gaussian inference problem of
inferring the initial condition of a linear dynamical system from noisy output
measurements taken after the initial time. In practical applications, the large
dimension of the dynamical system state poses a computational obstacle to
computing the exact posterior distribution. Model reduction offers a variety of
computational tools that seek to reduce this computational burden. In
particular, balanced truncation is a system-theoretic approach to model
reduction which obtains an efficient reduced-dimension dynamical system by
projecting the system operators onto state directions which trade off the
reachability and observability of state directions as expressed through the
associated Gramians. We introduce Gramian definitions relevant to the inference
setting and propose a balanced truncation approach based on these inference
Gramians that yield a reduced dynamical system that can be used to cheaply
approximate the posterior mean and covariance. Our definitions exploit natural
connections between (i) the reachability Gramian and the prior covariance and
(ii) the observability Gramian and the Fisher information. The resulting
reduced model then inherits stability properties and error bounds from system
theoretic considerations, and in some settings yields an optimal posterior
covariance approximation. Numerical demonstrations on two benchmark problems in
model reduction show that our method can yield near-optimal posterior
covariance approximations with order-of-magnitude state dimension reduction.
