[gjm11]

You could, but you should follow it up with "where one of those sources is your (incomplete, noisy) information about what the value _used_ to be and how it changes over time".

If you just have (say) three different ways of estimating the position of an aeroplane, then you don't need a Kalman filter. The place where the Kalman filter adds value is where you measure its position _repeatedly_ and make use of the fact that it's known to be moving approximately in a straight line.

[bigred100]

How do EKF and ensemble Kalman Filters fit into this?

[gjm11]

The Kalman filter assumes that all errors are gaussian, that all updates are linear (i.e., new state = linear function of old state), and that observations are linear (i.e., what you measure is a noisy version of a linear function of the state).

The extended Kalman filter allows for state updates and observations to be nonlinear, by the straightforward expedient of replacing them with linear approximations near to the current estimate.

The ensemble Kalman filter also allows for nonlinear updates and observations; instead of keeping track of the expectation and covariance of the state (i.e., everything you need to define a Gaussian model, things that behave nicely under linear transformations but not under nonlinear ones) it keeps track of an "ensemble" of sample values, applies the update and observation functions to those, and then estimates expectations and covariances from this ensemble. It's a sort of Monte Carlo Kalman filter.

A couple of other things worth knowing about:

Intermediate between the extended KF and the ensemble KF is the "unscented Kalman filter". Like the ensemble KF it estimates things using a number of samples; but instead of propagating those samples through repeated steps, it picks the sample points at each step on the basis of the estimated expectation and covariance, and uses them only to compute new expectations and covariances. More expensive than the EKF but copes better with substantial nonlinearities.

Extrapolating beyond the ensemble KF is the "particle filter", which uses the same "track an ensemble of samples" approach but gives up the assumption that all the errors are Gaussian. You need a larger ensemble to get good results, I think, but it can cope with a wider range of scenarios. (I find the name "particle filter" annoyingly distracting; the "particles" are the samples, which I guess you're supposed to think of as a cloud of points in possible-configuration-of-the-system space, and of course it's a "filter" in the same way as the Kalman filter is.)