Bayesian Estimation and Tracking: A Practical Guide

3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance

Introducing a notation that will be used throughout the remainder of this text, let an estimate of x_n conditioned on all observations up to time t_p be written as with

(3.25)

Thus, an estimate of x_n that uses all observations, including the current one at time t_n, is written as

(3.26)

while one that predicts x_n based on all but the current observation is given by

(3.27)

The same can be done for the covariance matrix, writing

(3.28)

with prediction form of the covariance given by

(3.29)

3.4.1 State Vector Prediction

Using the Chapman–Kolmogorov equation (3.24) in (3.27) and (3.29) results in

(3.30)

and

(3.31)

From the system dynamic equation (3.17), (3.30) can be rewritten as

(3.32)

(3.33)

In moving from ...