Environmental Engineering Reference
In-Depth Information
The goal is now to quantify the impact of the measurement outcome
Z
on the parameters
X
and, ultimately, on the event of interest
F.
In Bayesian analysis, this is carried out by
computing the conditional probability of
F
given the information
Z:
Pr(
F|Z
)
.
For the above
example, we assume that the measured deformation is more than initially predicted. In this
case, the new information will increase the estimated probability of
F,
that is,
Pr(
F|Z
)
>
Pr(
F
)
.
The conditional probability is defined as
Pr(
FZ
Z
∩
)
(5.1)
Pr(
FZ
|
)
=
.
Pr()
Following Bayes' rule, this conditional probability can be calculated by
Pr(
ZF
)Pr()
Pr()
|
F
(5.2)
Pr(
FZ
|
)
=
.
Z
In Bayes' rule (and in Bayesian analysis, in general), the terms have the following meaning:
Pr(
F
): prior probability;
Pr(
F|Z
): posterior probability;
Pr(
Z|F
): likelihood;
Pr(
Z
): probability of making the observation.
Bayes' rule tells us how a prior probability Pr(
F
) is updated to a posterior probability
Pr(
F|Z
) when making the observation
Z.
The observation is described by the likelihood
term Pr(
Z|F
), which is introduced in more detail in Section 5.3.3.
In many instances, one is not only interested in a single event
F
, but in the distribution of
all parameters
X
. Bayes' rule can be extended to random variables
X
:
f
X
x
′′
()
=
aL
() ().
X
x
f
′
x
(5.3)
where ′
f
X
is the prior probability density function (PDF) and ′′
f
X
is the posterior PDF, that is,
X
|
is the conditional PDF of
X
given the observation
Z. L
(
x
) is the likelihood func-
tion, which, in analogy to
Equation 5.2
,
is defined as
L
f
′′
=
f
Z
()
x
∝
Pr(
Z
|
X
=
x
);
here, the propor-
tionality is with respect to
x
(see Section 5.3.3). The constant
a
is
1
a
=
.
∞
(5.4)
∫
Lf
() ()
xxx
X
′
d
−∞
∞
∞
…
x
∫
∫
∫
Here, we use the convention
d
x
=
d
…
d
.
a
is often called the normalizing
1
n
X
−∞
−∞
constant, since it ensures that
can be observed that
a
corresponds to 1/Pr(
Z
), that is, it is an indication for how likely the
observation is.
The application of Bayes' rule for updating a single random variable is presented in the
following illustration.
′′
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