Kalman Filter - Order of Update Step?
If you look at Gary Bishop, Greg Welch - An Introduction to the Kalman Filter (SIGGRAPH 2001, Course 8) then, on page 17, there are equations 3.3 and 3.4:



This is probably not the KF formulation you normally see because the system equation, 3.3, is kind of lagged one behind the observation equation, 3.4, time-wise ( with respect to their right hand sides ). This is less common than the one where the RHS's of both equations has the same time subscript. So, without going into all the gory details, if you derive the KF based on 3.3 and 3.4, you get totally different recursions but they are equivalent to those in the KF formulation where the RHS's of the observation and system equations are contemporaneous rather than lagged by one. Hopefully this gives the general idea. I would have put it in comments since it doesn't provide the details but it wouldn't fit.