Deriving the Discrete-Time Kalman Filter from Finite-Horizon Batch Estimation

The Kalman filter is typically presented as a sequence of Bayesian conditioning steps. While compact, that viewpoint can obscure where the recursion comes from algebraically and what global object it is implicitly factorizing. This note makes that structure explicit by centering the derivation on a single finite-horizon quantity: the inverse stacked innovation covariance G_1:k. Viewing filtering through G_1:k clarifies how innovations are coupled across time by the dynamics, how conditioning and sensitivity appear in innovation space, and how fixed-window and smoothing variants correspond to retaining more of the same batch structure.

Concretely, state estimation over 1:k is posed as a constrained quadratic program in the stacked process disturbances and measurement noises. Eliminating the primal variables in the KKT system yields the inverse batch innovation matrix shown below.

Extending the horizon from k−1 to k induces a structured block recursion for G_1:k; the Schur complement of the appended block is exactly the per-step innovation covariance S_k, and the Kalman gain appears as the corresponding block factor. Gaussianity is required only to interpret the quadratic program as a MAP estimator; all subsequent steps are deterministic linear algebra.

Written for readers already comfortable with linear systems, numerical linear algebra, and constrained optimization; it is not an introductory treatment of Kalman filtering.