Environmental Engineering Reference
In-Depth Information
The posterior parameters are calculated as
n
λ
′′+
′ +
ν
n
n
ϕ
λλ αα ββ ϕϕ
n
(
)
2
n
λ
(
ϕ
− ′
ν 2
)
i
ν
′′ =
,
′′ = ′ +
n
,
′′ =
′ +
,
′′ = ′ +
+
.
λ
2
2
λ
′ +
n
2
i
=
1
A noninformative prior is obtained by choosing λ′ = 0, α′ = −(1/2), and β′ = 0.
We consider three direct shear tests (as in Illustration 1), resulting in friction angles
φ 1 = 25.6°, φ 2 = 25.5°, φ 3 = 24°. Inserting the non-informative prior parameters and these
samples into the above expressions, we obtain the posterior parameters as
ν
′′ =
25 03
.
°
,
λ
′′ =
3
,
α
′′ =
1
,
β
′′ =
0 803
.
.
ϕ µ( , is shown in Figure 5.6 , together
with the posterior distribution obtained when assuming that the standard deviation is fixed
at σ φ = 3° (also with a noninformative prior, as calculated in Illustration 3). Note that in this
case, the assumption of an uncertain standard deviation of φ leads to a lower standard devia-
tion in the estimate of μ φ . However, the tail of
The resulting marginal distribution of the mean, ′′
f µϕ
ϕ µ() is heavier in this case, as is evident when
plotting the PDF in log scale (right-hand side of Figure 5.6 ) . As an example, the probability
Pr(μ φ ≤ 20°) is 2 × 10 −3 in the case of σ φ = 3° and 4 × 10 −3 in the case of an uncertain σ φ .
f µϕ
′′
5.3.4.2 Numerical integration to determine the proportionality constant
A possible solution to the Bayesian updating problem is the numerical evaluation of the
proportionality constant a
[
]
X d 1 If the number of random variables in X is
small, classical integration schemes based on quadrature rules are applicable. The full pos-
terior PDF is then available through Equation 5.3 .
For larger numbers of random variables, it is still possible to evaluate a approximately
using sampling schemes (see Section 5.4). However, making predictions with the posterior
distribution also requires integration over X . If a sampling method is used to determine a ,
it is therefore more convenient to directly work with these samples when doing predictions
=∫
L
() ()
xxx
f
.
−∞
(a)
0.7
(b)
10 0
0.6
Variance σ φ 2
uncertain
10 −2
0.5
Variance σ φ 2
uncertain
0.4
10 −4
0.3
Variance σ φ 2
fixed
0.2
Variance σ φ 2
fixed
10 −6
0.1
0
10 −8
15
20
25
30
35
15
20
25
30
35
μ φ (°)
μ φ (°)
Figure 5.6 Posterior distributions of the mean friction angle, once with uncertain mean and standard
deviation, and once with uncertain mean and fixed standard deviation σ°
ϕ = 3 . (a) Normal scale;
(b) log scale.
 
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