Assessing the value of information to design site investigation and construction quality assurance programs - Risk and Reliability in Geotechnical Engineering - page 491

∑
all θ

PDecision ConsequenceInformation ( ) = PConsequen ( ce θ , … , θ ΘΘ )( , P … , ε )

1 n 1 n

,, … θ

1 n

(13.11)

When working with model parameters, it can be insightful to express the likelihood func-
tion in terms of the decision consequences (e.g., Figure 13.11 ) since it is the relationship
between decision consequences and information that governs the value of information:

PInformation Decision Consequence ( )

∑
all θ

∑

= PConsequen ( ce θ , … , θ ΘΘ Εθ θΘ Θ )( , P … , ε ) × P ( = ε , … , )( , P … , )

1 n 1 n 1 n 1 n

all

,, … θ

1 n

θ ,, … θ

1 n

∑

PConsequence ( θ , … , θ ΘΘ )(,, P … )

(13.12)

1 n 1 n

all θ ,, … θ

1 n

13.3.3 Illustrative example: Design of pile foundation
An example illustrating insights from Bayes' Theorem is for the design of a pile foundation
for axial loading ( Figure 13.12 ). The decision is between two different pile lengths. Spatial
variability in axial pile capacities across the site is modeled with a normal distribution,

Construction
cost
(per pile)

Failure
cost
(per pile)

R = Capacity
S = Load
R > S

Θ = Gradient of average unit
side shear versus depth

$4500 $0

A: 30 m pile

P (Θ)

Do not test any
pile

E ( C ) = $6870

R < S $4500 $45,000

R > S $6125 $0

B: 35 m pile

P (Θ)

E ( C ) = $6550

R < S $6125 $61,250

R > S $4500 $0

30 m pile

P (Θ| )

R < S $4500 $45,000

R > S $6125 $0

Te st a number
of piles

35 m pile

P (Θ| )

R < S $6125 $61,250

.

Uncertain test
pile capacities
p E ( )

R > S $4500 $0

30 m pile

P (Θ| )

R < S $4500 $45,000

R > S $6125 $0

35 m pile

P (Θ| )

R < S $6125 $61,250

Figure 13.12 Decision tree to assess value of information for load tests in pile foundation design.