## Monday, May 17, 2004

### Likelihood function and Bayes' theorem

"Bayesian Multiple Target Tracking"
@book{Stone99Bayesian,
Year = {1999},
Title = {Bayesian Multiple Target Tracking },
Isbn = {1580530249 },
Booktitle = {Bayesian Multiple Target Tracking },
Publisher = {Artech House},
Author = {Lawrence D. Stone, and Carl A. Barlow, and Thomas L. Corwin}}

Likelihood functions replace and generalize the notion of contact used in linear Gaussian trakers. For this reason, likelihood functions play a key role in Bayesian tracking. The likelihood function L for the random variable X and observation Y=y is defined to be: $L(y|x) = Pr{Y=y | X=x} for x\in S$. Baye' theory: $p(x|y) = \frac{L(y|x)p(x)}{\int L(y|x)p(x)dx}$. The distribution given by p is called the prior distribution and the distribution given by $p(.|y)$ is called the posterior distribution on X given the observation Y=y. There are several typical likelihood functions such as: Gaussian contact model, Gaussian bearing-error model, Single-plus-noise model.