The variance of the combined Gaussian will not always shrink with more data. Consider the case that you have one sensor that predicts 100m at a high accuracy and then you get another sensor predicting 10,000m with the same accuracy. At first you thought you had a very good idea of the location then with more information you realize that you actually have no idea and you get a Gaussian with mean of 5,050m and a very high variance.
The post left out the some important parts of the Kalman Filter, where you adjust your certainty in how well you're able to predict where the target will go and how accurate your sensors are in response to how they perform over time. So even if you're Gaussians are 10 klicks apart the variance for this prediction still shrinks, but your future Gaussians would be much wider. I think what was described in the post was more an alpha-beta filter.
Adjusting the measurement covariance to fit the residuals is not part of the Kalman filter algorithm. One could make the argument that any real-world implementation needs to address this problem, but as far as the algorithm is concerned, the measurement covariance is externally computed.
The sensors cannot both have the high accuracy because one or both of them are clearly wrong in your scenario, meaning the estimated accuracy of one or both sensors is clearly wrong.
If your estimate of the accuracy of the sensor is wrong, you are giving a very high accuracy weight to the garbage going in and you will get garbage out.
One of the really interesting things about a Kalman filter is that it improves its estimate of the accuracy of the input sensors over time. In your scenario, over time the Kalman filter would "learn" which sensor is lying and adjust its accuracy estimate down.
> over time the Kalman filter would "learn" which sensor is lying and adjust its accuracy estimate down
No, it won't. The plain old Kalman filter believes precisely what you tell it. In this scenario, the Kalman filter will oscillate about the mean of the two measurements, with the amplitude of that oscillation depending on the relative sizes of the measurement variance and estimate variance. A smaller process noise will cause the estimate covariance to shrink faster, which will dampen the oscillation faster. The oscillation will eventually settle on some minimum amplitude.
An important point here that many don't realize is that the covariance of the Kalman filter is completely independent of the residual. Go ahead, look at the equations - covariance is a function of the measurement model, the prior covariance, and the measurement covariance. The actual measurement doesn't matter.
No, it does always shrink. It's counter-intuitive, but that's what the formula gives. What you're pointing out is that some situations aren't really that well modeled by a Gaussian and you probably want something with a thicker tail.
