Assessing the value of information to design site investigation and construction quality assurance programs - Risk and Reliability in Geotechnical Engineering - page 487

This example of the contaminated lagoon is loosely based on an actual case history.
In the case history, the pilot test produced a successful result and in situ bioremediation
was selected and implemented. While this approach was effective in essentially removing
the source of contamination from the lagoon, it was not successful in removing all of the
contamination in the ground at the site because pools of non-aqueous-phase-contaminant
liquids had migrated from the lagoon. Consequently, the conventional pump-and-treat
approach was still required in the long term to contain groundwater contamination on-
site, resulting in a significantly greater cost (i.e., the event F B occurred when Alternative B
was selected in Figure 13.3 ). Therefore, this case history underscores the important role of
uncertainty. A preferred alternative may have the greatest expected value, but the actual
outcome when the decision is implemented may not be preferred. In addition, the value of
information is an expected value and not necessarily the actual value that will be realized
by obtaining additional information.

13.3 InSIghtS FroM baYeS' theoreM

Bayes' Theorem ( Equation 13.5 ) plays an integral role in the value of information. Bayes'
Theorem can be expressed as follows:

PDecision ConsequenceInformation
PInformation Decision Con

( )

= ( sequencePDecision Consequence )( )

(13.7)

∑

PInformat ( ionDecision ConsequencePDecision Consequence )( )

all
Consequences

where PDecision ConsequenceInformatio ( ) is the updated probability for the possible deci-
sion consequence given the information, P ( Decision consequence ) is the prior probability
for the consequence, and PInformation Decision Consequenc ( ) is the likelihood function
relating the probability of obtaining the information if that decision consequence is realized.
The two key components in Bayes' Theorem are the prior probability and the likelihood
function ; the set of prior probabilities provides the starting point and the likelihood func-
tion serves to filter the starting point based on the information.

13.3.1 Prior probabilities
The prior probabilities in Bayes' Theorem are important because the updated probability
for a decision consequence is proportional to the prior probability for that consequence
( Equation 13.7 ). The importance of the prior probabilities is underscored by considering
the case where the prior probability for a particular decision consequence is zero; no matter
how strong the likelihood function is in amplifying the probability of that consequence, the
updated probability will always be zero.
The prior probabilities establish the initial uncertainty in the decision consequences. A
common misconception is that additional information will reduce uncertainty; uncertainty
can decrease, remain unchanged, or increase with additional information. Consider the
simple example shown in Figure 13.10 . While the prior probability distribution is heavily
weighted toward a consequence of $10 MM with little uncertainty ( Figure 13.10a ), the