The Bayes' rule allows us to modify our knowledge about a system/process using both the historical information and the current data according to the following rule:
where the density function 
characterizes
the prior information about
the parameters 
of the process,
represents the likelihood of observing the
data 
given 
, and 
is the updated
knowledge about 
based on both the prior and the sensed data.
The involved parameters 
are often estimated by minimizing a
risk function which depends on a
prespecified loss function and the posterior distribution.
A commonly used 0-1 loss function results in an estimate of 
which maximizes the posteriori distribution 
,
which is the Maximum A Posteriori (MAP) solution.
