Applying Kalman filter to a data set

I went through the answer Kalman filter in practice and it seems we must know all the first and second order properties of random variables to apply the Kalman filter.

But when I only have a set of data $y_1,\dots,y_n$ and a given model $G$ and $F$ with assumed noise processes, could someone please explain to me how to apply Kalman filter?

Thanks.