I am not sure but I think that Airbus FMS does not use Kalman filter in that way.
It chooses the most accurate and reliable navigation source depending on the sensors available, geometric dilution of precision, altitude, etc…
The MIX IRS position uses a weighed average of the three IRS positions, giving more weight to those IRS with historically better precision as compared with reliable radio or GPS position.
GPIRS position is an input coming from the ADIRU. It is not FM computed! Each ADIRU uses its associated GPS and gives a position with good accuracy (GPS) augmented with the ability of the IRS to sense acceleration and speed. (GPS is better if it remains still). It is even better than pure GPS position, basically. I don't know with mathematical method is used for that mix.
So when the FMS decides that GPS is reliable it will simply use GPIRS position.
If GPS is not reliable or available (or even installed), then the position will be either radio position (DME/DME or VOR/DME, or LOC DME DME) when this source is considered as the best one, or IRS if radio is not (low altitude just after take off, bad dilution of position, no navaids...). I think there is some blending of these two (radio and IRS) when there is a transition from one source or the other, this blend tending to one or the other. Perhaps it is carried out by those Kalman filters.
The BIAS vector is like the correction you make to your gun when you shoot at the target. If all your shoots go up and to the right by 5 inches, then shoot low and to the left by five inches and you will hit the target. If accurate and reliable radio position is 2 miles north and 1 mile east of IRS position and then you lose radio sensors, then the FMS will give an position which is 2 miles south and 1 mile west of the IRS position, instead of simply giving pure raw IRS position.
hope it helped