Editing Expectation–maximization algorithm (section)

== 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:

;E-step
: Operate a Kalman filter or a minimum-variance smoother designed with current parameter estimates to obtain updated state estimates.

;M-step
: Use the filtered or smoothed state estimates within maximum-likelihood calculations to obtain updated parameter estimates.

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
: <math>\widehat{\sigma}^2_v = \frac{1}{N} \sum_{k=1}^N {(z_k-\widehat{x}_k)}^2,</math>

where <math>\widehat{x}_k</math> are scalar output estimates calculated by a filter or a smoother from N scalar measurements <math>z_k</math>. 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
: <math>\widehat{\sigma}^2_w = \frac{1}{N} \sum_{k=1}^N {(\widehat{x}_{k+1}-\widehat{F}\widehat{{x}}_k)}^2,</math>

where <math>\widehat{x}_k</math> and <math>\widehat{x}_{k+1}</math> are scalar state estimates calculated by a filter or a smoother. The updated model coefficient estimate is obtained via
: <math>\widehat{F} = \frac{\sum_{k=1}^N {(\widehat{x}_{k+1}-\widehat{F} \widehat{x}_k)}^2}{\sum_{k=1}^N \widehat{x}_k^2}.</math>

The convergence of parameter estimates such as those above are well studied.<ref>{{Cite journal
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