Measurement Update Derivations

Minimum Mean Square Error (MMSE) Estimation

Consider the following static state estimation problem: Given a prior distribution (probability density function or pdf) for a Gaussian random vector with dimension of and a new dimentional measurement corrupted by zero-mean white Gaussian noise independent of state, , we want to compute the first two (central) moments of the posterior pdf . Generally (given a nonlinear measurement model), we approximate the posterior pdf as: . By design, this is the (approximate) solution to the MMSE estimation problem [Kay 1993] [24].

Conditional Probability Distribution

To this end, we employ the Bayes Rule:

In general, this conditional pdf cannot be computed analytically without imposing simplifying assumptions. For the problem at hand, we first approximate (if indeed) , and then have the following joint Gaussian pdf (noting that joint of Gaussian pdfs is Gaussian):

Substitution of these two Gaussians into the first equation yields the following conditional Gaussian pdf:

We now derive the conditional mean and covariance can be computed as follows: First we simplify the denominator term in order to find the conditional covariance.

where we assumed is invertible and employed the determinant property of Schur complement. Thus, we have:

Next, by defining the error states , , , and using the matrix inersion lemma, we rewrite the exponential term as follows:

where since covariance matrices are symmetric. Up to this point, we can now construct the conditional Gaussian pdf as follows:

which results in the following conditional mean and covariance we were seeking:

These are the fundamental equations for (linear) state estimation.

Linear Measurement Update

As a special case, we consider a simple linear measurement model to illustrate the linear MMSE estimator:

With this, we can derive the covariance and cross-correlation matrices as follows:

where is the discrete measurement noise matrix, is the measurement Jacobian mapping the state into the measurement domain, and is the current state covariance.

where we have employed the fact that the noise is independent of the state. Substitution of these quantities into the fundamental equation leads to the following update equations:

These are essentially the Kalman filter (or linear MMSE) update equations.