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Exact filtering for the declared linear Gaussian model
Statement
Assume AC. Let and take an admissible stochastic coefficient model from the definition of the finite coefficient observation and innovation interface. Thus the coefficient states and observations satisfy and , the initial state and all process and measurement noises are mutually independent Gaussian vectors with positive-semidefinite covariances, and every innovation covariance used by the recursion is positive definite. Run the finite-horizon Kalman recursion from the declared initial mean and covariance. Then, for each , its Gaussian output is an everywhere-defined conditional-law version for the coefficient state given the observation history . The Gaussian mean and covariance at each history are those of this chosen version. These values are defined even on sets of histories having zero probability under the model, although another version need not agree there. Initial, process, measurement, forecast-state, and analysis-state covariances may be singular; only the innovation covariances being solved must be positive definite. This is a deduction about the declared finite model, not a claim that a nonlinear or untruncated fluid equation is itself linear-Gaussian.
Statement references
Facts & Assumptions
Given: AC, a finite-horizon stochastic coefficient model from the definition above, and positive definiteness of every innovation covariance that is solved.
Assume AC, the axiom that every family of nonempty sets has a choice function. The Axiom of Choice
The local coefficient realization has the affine state and observation equations, mutually independent Gaussian initial, process, and measurement vectors, and positive-semidefinite covariance matrices. Finite coefficient observation and innovation interface
This is the finite linear-Gaussian state-space model with independent noises; its state and observation dimensions and matrices satisfy the imported model definition. Linear-Gaussian state-space models with independent noises
Under AC and these hypotheses, the finite-horizon Kalman recursion gives an everywhere-defined filtering kernel at every observation history; singular state and noise covariances are allowed when each solved innovation covariance is positive definite. The finite-horizon linear-Gaussian Kalman filter is exact
Proof
Given: The hypotheses in the Statement.
Identify the coefficient state with the state vector in the finite linear-Gaussian state-space model: the local matrices are and , and the local Gaussian vectors are the initial state and the mutually independent . Their dimensions and covariance signs match the supplied model by [F1] and [F2]. The stated positive-definite condition is exactly the admissibility condition for each covariance solve, and [A1] matches the choice assumption of the Gaussian suppliers. Thus all hypotheses of [F3] hold.
Apply [F3] to the same coefficient state, fixed observation history, and Kalman recursion: for every from through , the computed Gaussian law is a conditional law of given , with the exact conditional mean and covariance stated above. The theorem allows singular state, process, and measurement covariances, so no inverse beyond the positive-definite innovation matrices is required.