Inference and Learning from Data Inference

In this chapter we illustrate one important application of the linear meansquare-error (MSE) theory to the derivation of the famed Kalman filter. The filter is a powerful recursive technique for updating the estimates of the state (hidden) variables of a state-space model from noisy observations. The state evolution satisfies a Markovian property in the sense that the distribution of the state xn at time n is only dependent on the most recent past state, xn−1. Likewise, the distribution of the observation yn at the same time instant is only dependent on the state xn. The state and observation variables are represented by a linear state-space model, which will be shown to enable a powerful recursive solution. One key step in the argument is the introduction of the innovations process and the exploitation to great effect of the principle of orthogonality. In Chapter 35 we will allow for nonlinear state-space models and derive the class of particle filters by relying instead on the concept of sequential importance sampling.