
8 Modeling and Inverse Problems in the Presence of Uncertainty
and
Prob{A
k
|B} =
Prob{A
k
, B}
Prob{B}
=
Prob{B|A
k
}Prob{A
k
}
P
l
j=1
Prob{B|A
j
}Prob{A
j
}
. (2.9)
Equation (2.8) is often called the law of total probability. Equation (2.9) is
known as Bayes’ formula or Bayes’ Theorem. Here Prob{A
k
} is called a prior
probability of event A
k
, Prob{B|A
k
} is called the likelihood of B given A
k
, and
Prob{A
k
|B} is called a posterior probability of event A
k
obtained by using
the information g ained from B.
2.2 Random Variables and Their Associated Distribu-
tion Functions
In most applications of probability theory, we are not interested in the
details associated with each sample ...