The last five years have witnessed a really significant increase in the awa
reness of numerical Bayesian methods, both in Statistics and in Signal Proc
essing. It is now clear that many problems that could only be addressed usi
ng ad hoc methods, because of their complexity, can now be solved and these
solutions can be applied to almost all areas of data and signal processing
. Bayesian methods have been popular for decades. However, various approxim
ations have been required in order to make progress because most of the int
egrations required within the framework have no analytical solutions apart
from some simple models which usually involve Gaussian and linearity assump
tions. This explains why sub-optimal, ad hoc approximations have been devel
oped. The aim of this paper is to set out the foundations upon which modern
numerical Bayesian methods are based, give one application to missing data
in audio restoration and then give references to application areas that ca
n be addressed. (C) 2001 Elsevier Science B.V. All rights reserved.