the context of my questions is the following:
assuming a uniformly distributed random variable ( defined by physics in [a; b]) as a prior I want to identify the correct value of it via the Kalman Filter. As far as I read the KF is only applicable to normally distributed RVs. So my approach would be to transform the uniform RV to a normal RV, do the update and retransform it to a uniform distribution.
First of all, it this approach correct?
Second, if the approach is correct, how do I perform the transformation? All I found was the Box-Muller method, but this method needs two initial uniform RVs. I guess I can somehow just assume a certain 2nd RV to satisfy this condition. However I am not sure how this 2nd dummy RV needs to be defined so that it doesn't break the physical conditions the 1st RV (needs to) satisfies.
Sometimes handcrafting is necessary to make things work. In my personal experience, I used many times KF with uniform noises and it worked.
From a theoretical point of view, if you use something different from gaussian noise, you are not guaranteed that the KF is optimal (i.e. the estimation may be suboptimal). But anyway, it may still work good.