Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 39

(absence of other sources of information). Among the unconditional ( s u , N) samples, there
are some samples satisfying N ≈ 10 (9.9 < N < 10.1). These samples are marked as dark
gray crosses in the left plot. Similarly, the ( s u , q t - σ v ) samples satisfying q t − σ v ≈ 2000 kPa
(1990 < q t − σ v < 2010) are marked in dark gray in the right plot. Finally, the samples satis-
fying N ≈ 10 and q t − σ v ≈ 2000 kPa are marked as black triangles in both plots. These are
the conditional samples of s u given the information N ≈ 10 and q t − σ v ≈ 2000 kPa.
Figure 1.17 shows the histograms of the unconditional and conditional s u samples. There
are two distinct features: (a) the mean value of s u changes after the conditioning, and (b)
the COV of s u is smaller after the conditioning. This illustrates the effect of conditioning:
given the information that N ≈ 10 and q t − σ v ≈ 2000 kPa, the probability distribution can
be updated. The updated (conditional) distribution has a mean value different from the
unconditional mean and has a COV that is less than the unconditional COV. The reduc-
tion in COV deserves more attention: it means that the magnitude of uncertainty is reduced
in the presence of additional sources of information. Engineers can appreciate this effect
intuitively, but conditioning allows this effect to be quantified consistently. The effect of
conditioning on the distribution of

σ p is similar.
The above example is intended to illustrate the effect of conditioning without mathe-
matics. In fact, the conditional distribution of (, )

′

s u ′ σ can be obtained analytically using
Bayesian analysis. The tedious simulation involving 2 million samples can be avoided. The
information N = 10 and q t − σ v = 2000 kPa is equivalent to

( ) =− ( ) =−

XN =− µσ 10 15 6 . 11

3 3 3

(1.74)

( ) − ( )

X = q t − σµσ = 2000 − 2500 7 50 =−. 067

4 v 4 4

Let X be partitioned in X [1] and X [2] , where X [2] = [X 3 , X 4 ] T = [−1.1 −0.67] T are known
values, and X [1] = [X 1 , X 2 ] T are unknown random variables:

=

X
X

[1]

X (1.75)

[] 2

× 10 5

5 80

70

Unconditional s u samples Conditional s u samples

4

60

Unconditional
distribution
of s u

Conditional
distribution
of s u

50

3

40

2

30

20

1

10

0 0

0

100 200

s u (kPa)

300 400 0 100 200

s u (kPa)

300 400

Figure 1.17 Unconditional and conditional PDFs of s u .