6.3 The Linear Kalman Filter
When both the dynamic and observation processes are linear, Bayesian estimation reduces to the LKF. Note that it can be shown that the prediction equations can be derived directly from (5.3) to (5.7) using general distributions for the posterior and prior densities, making the linear Kalman filter applicable for any density. Only the requirements of linearity of the dynamic and observation equations need to be applied. However, this requires the density function to be of known analytical form so that the moment integrals can be evaluated in some manner. In many cases this has proven to be an insurmountable requirement.
The complete LKF process is presented in Table 6.1.
Table 6.1 Linear Kalman Filter Process.
| Step 1. | Filter initialization: | Initialize  and  |
| Step 2. | State vector prediction: |  |
 |
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| Step 3. | Observation-related prediction: |  |
 |
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