[jampekka]

I think the easiest way depends on your background knowledge. If you understand linearity of the Gaussian distribution and the Bayesian posterior of Gaussians, the Kalman filter is almost trivial.

For (1D) we get the prior from the linear prediction X'1 = X0*a + b, for which mean(X'1) = mean(X0)*a + b and var(X'1) = var(X0)*a^2, where a and b give the assumed dynamics.

The posterior for Gaussians is the precision weighted mean of the prior and the observation: X1 = (1 - K)*X'1 + Y*K, where the weighting K = (1/var(X'1))/(1/var(X'1) + 1/var(Y)), with Y being the Gaussian observation.

Iterating this gives the Kalman filter. Generalizing this to multiple dimensions is straightforward given the linearity of multidimensional Gaussians.

This is how (after I understood it) it makes it really simple to me, but things like linearity of (multidimensional) Gaussians and the posterior of Gaussians as such probably are not.

[krtab]

I have written down a similar derivation here if anyone is interested: https://ngr.yt/blog/kalman/

[RossBencina]

What you write is simple. But your scalar model suppresses the common situation of a measurement matrix with output dimension less than state dimension. Exactly how the Kalman gain formula works under this setting I'm less clear on. Beyond that, additional insight is needed when the measurement matrix is non-linear and K = P_xy P_y^{-1} as in the UKF. At least I get stuck there, with little formal statistics work.

[jampekka]

Good catch, indeed a measurement matrix is needed if the state and measurement are of different dimensions or require a (linear) transformation. For that use Y = H*z where H is the measurement matrix and z is the observation vector.

For UKF the Y is still a multidimensional Gaussian and computing K is the same. The mean and covariance of Y is computed from Z and the nonlinear measurement function using the unscented transform.