I have a large number of of noisy time series recordings (trials), for which I wish to estimate the state transition model underlying them using the Kalman filter. The process generating the time series is the same in every trial, and so I want to be able to pool information from all of the trials, rather than fitting individual filters to every trial.
I thought this should be a fairly common application of the Kalman filter, but unfortunately I haven't been able to find any literature on how this might be done. I suspect I've either fundamentally misunderstood something about how Kalman filters work, and so what I require can't be done, or I just don't know the terminology for what I'm looking for.
So, can anyone tell me if it is possible to fit the Kalman filter in this way, and if so, what this method is called?