Georgia Institute of Technology College of Sciences

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 control-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.  We define an analogous balanced truncation procedure for the Bayesian inference setting based on the trade off between prior uncertainty and data information. The resulting reduced model inherits desirable theoretical properties for both the control and inference settings: numerical demonstrations on two benchmark problems show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.