14
Decisions in Uncertain Contexts
This chapter explores how to m ake rational deci si ons regar di n g
actions to be t aken in a context where the state variables on which
a dec i si on is based are themselves uncer t ain . Basic contextual
notions will be presented through an illustrative example before
introducing the formal mathematical notation associated with
rational decisions and value of information.
14.1 Introductory Example
(a) Given an
m
3
of possibly
contaminated soil.
Pr = 0.9
Pr = 0.1
$0
9$100
9$10K
9$100
E[$| ]=9$1K
E[$| ]=9$100
(b) State probabilities, values, and
expected values.
Figure 14.1: Decision context for a soil
contamination example.
In or de r to introdu ce the noti ons associat ed wit h utility theory,
we re vi s it the soil contamination example presented in
§
8.2.5,
where we have a cubic meter of potentially contaminated soil . The
two possi b l e states
x 2{, }
are e i t he r contaminated or not
contaminated, and the two possi bl e acti ons
a 2{, }
are e i t he r to
do nothing or t o send the soil to a recycling plant where it is going
to be decontaminated. A soi l sampl e is deﬁned as contaminated
if the pollutant concentration
c
exceeds a threshold value
c>
adm.
. Th e issu e is that for most m
3
in an indu st r i al sit e, the re
are n o observations made in order to verify whether or not it is
contaminated. Here, we rely on a regre ssi on model buil t from a set
of d is cr et e obser vations to predi ct for any
x, y, z
coordinates what
our knowledge of t he contaminant conc entration is as descri bed
by it s expected value and variance. In this example, our current
knowledge indicates that, for a speciﬁc m
3
,
Pr(C
adm.
) ⌘ Pr(X = )=0.9
Pr(C>
adm.
) ⌘ Pr(X = )=0.1.
The optimal decision regarding the action to choose depends
on t he value incurr ed by taking an action given the state
x
.In
this example we have two states and two actions, so there are four
possibilities to be deﬁned, as illustrate d in ﬁgure 14.1. The incurred