A linear Gaussian state space model can be written in the following form (Durbin and Koopman, 2012):

The measurement or observation equation with for is given by

.

The state equation (transition dynamics) with , for is given by

,

with initialisaiton

.

Note:

- The system matrices , , , , , and are predetermined matrices. Usually they are time-invariant and functions of unknown parameters.
- Initialisation oftentimes plays an important role in inference. For non-stationary components in , say , conventionally diffuse initialisation is used. In such a case,

and is diagonal with being a large number. For stationary components in , say , unconditional distribution is used to initialise. It follows,

and is such that

,

where is the dimension of stationary components in .

The Kalman filter computes and via a forward recursion. Prediction errors are produced as a by product. It follows

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