Smoothing estimation of stochastic processes: Change of initial condition formulas

By posing the change of initial condition (CIC) problem in the theory of smoothing in linear estimation in a setting stripped of all inessentials, simple, insightful derivations of most CIC formulas (and a new likelihood formula) are provided. Specifically, the CIC or partitioning problem is one of low rank perturbation to a covariance kernel and the formulas are simple consequences of inversion formulas for fixed rank modification of a positive definite kernel or matrix. The present derivation basically handles the discrete, continuous-discrete, and continuous cases at once: previous derivations had treated the discrete and continuous separately. The continuous-discrete results are new.