Expectation–maximization algorithm
Filtering and smoothing EM algorithms

A Kalman filter is typically used for on-line state estimation and a minimum-variance smoother may be employed for off-line or batch state estimation. However, these minimum-variance solutions require estimates of the state-space model parameters. EM algorithms can be used for solving joint state and parameter estimation problems.

Filtering and smoothing EM algorithms arise by repeating this two-step procedure:

Suppose that a Kalman filter or minimum-variance smoother operates on measurements of a single-input-single-output system that possess additive white noise. An updated measurement noise variance estimate can be obtained from the maximum likelihood calculation

where x ^ k {\displaystyle {\widehat {x}}_{k}} are scalar output estimates calculated by a filter or a smoother from N scalar measurements z k {\displaystyle z_{k}}. The above update can also be applied to updating a Poisson measurement noise intensity. Similarly, for a first-order auto-regressive process, an updated process noise variance estimate can be calculated by

where x ^ k {\displaystyle {\widehat {x}}_{k}} and x ^ k + 1 {\displaystyle {\widehat {x}}_{k+1}} are scalar state estimates calculated by a filter or a smoother. The updated model coefficient estimate is obtained via

The convergence of parameter estimates such as those above are well studied.
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