[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.