**Lesson 25 Maximum-likelihood State and Parameter Estimation**

**Summary**

The purpose of this lesson is to study the problem of obtaining maximum-likelihood estimates of a collection of parameters that appear in our basic state-variable model. A similar problem was studied in Lesson 11; there, state vector x(t) was deterministic, and the only source of uncertainty was the measurement noise. Here, state vector x(t) is random and measurement noise is present as well.

First, we develop a formula for the log-likelihood function for the basic state-variable model. It is in terms of the innovations process; hence, the Kalman filter acts as a constraint that is associated with the computation of the log-likelihood function for the basic state-variable model. ...