State space models and Kalman smoothing

Use the scalar model x_t=a x at t−1 plus w_t and y_t=h x_t plus v_t. The initial state is Gaussian with known mean and nonnegative variance. Process noises are independent centered Gaussian variables with variance q greater than zero; measurement noises have variance r greater than zero. These noise sequences and the initial state are mutually independent. Parameters are treated as known. These conditions make the conditional distributions Gaussian and ensure the denominators used below are positive.

Suppose the previous filtering mean and variance are m and P. Prediction gives a times m and a-squared times P plus q. The new measurement residual is its observed value minus h times the predicted mean, with variance S=h-squared times predicted variance plus r. Gaussian conditioning yields gain K equal to predicted variance times h divided by S. Updating adds K times the residual to the mean and reduces uncertainty according to the displayed variance formula.

After filtering through a terminal time T, the Rauch–Tung–Striebel smoother proceeds backward. It starts from the final filtering distribution. The gain J at t equals the filtered variance times a divided by the next predicted variance. The next smoothed mean differs from its earlier prediction because later observations supplied additional evidence. Multiplying that discrepancy by J transfers the appropriate correction to the earlier state. The covariance recursion transfers its associated reduction in uncertainty.

The smoothed state at t conditions on all measurements through T, including some after t. It is legitimate for reconstructing history once those measurements exist. A prediction issued at time t cannot use that later evidence. Conversely, smoothing entirely within a training record ending at a forecast origin is allowed, provided no post-origin data enter the fitted features. The information boundary is the last available observation, rather than the name of the numerical algorithm.

A future state forecast propagates the current filtering distribution without incorporating unobserved future measurements. Its variance accumulates process noise at each step. Forecasting a future measured value adds measurement variance as well. Those two variances answer different questions: uncertainty about an underlying signal is smaller than uncertainty about a fresh noisy measurement of it. Reporting only state uncertainty as an observation interval understates the latter.

Without joint Gaussianity, the same covariance calculations can describe best linear estimators under the corresponding second-moment and independence assumptions, but generally not the unrestricted conditional mean or its full distribution. Estimated parameters introduce uncertainty beyond the known-parameter filter covariance. Near-singular multivariate problems require numerical care and possibly alternative factorizations. A state space representation is a precise modeling choice; it does not by itself establish that a latent component is a physical cause or that the measurement errors match the assumed law.