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The goal of any estimator is to estimate one (or more) parameters of interest based on a model of the system, observations from sensors, or both. Because the parameters are, by definition, random vectors, they can be completely characterized by their associated probability density function (pdf). If we define our parameter vector and observation vectors at time k as x_k and z_k, respectively, the overarching objective of a recursive estimator is to estimate the pdf of all of the previous state vector epochs, conditioned on all observations received up to the current epoch. Mathematically, this is expressed as the following pdf:

(36.1)

where

(36.2)

and

(36.3)

While this is the most general case, it should be noted that most online algorithms would only be concerned with the conditional state estimate at the current epoch. For this situation, Eq. 36.1 would be represented as

(36.4)

In the next section, we will present the typical recursive estimation framework which will serve as the foundations for developing the forthcoming nonlinear recursive estimation strategies to follow.