Page 167 - Artificial Intelligence for the Internet of Everything

The Value of Information and the Internet of Things 153

The following theorem follows from Eq. (9.4).
Theorem 9.2 Using Assumptions 9.1 and 9.2, we have:
ZZ
EðVjbÞ¼ EðVjb,c,lÞ f ðcÞf ðlÞdc dl ðas aboveÞ
 2
Z ∞ Z ∞ (9.5)
f ðlÞdl f ðcÞdc
¼ ðb cÞ
∞ b
∞
Z
ðb cÞf ðcÞdc (9.6)
¼ PðL > bÞ
∞
(9.7)
¼ b EðCފ PðL > bÞ
½
The above corresponds to Eq. (9.10)(Howard, 1966). After our above
assumptions, to obtain EðVjbÞ, we only need the distribution of L and EðCÞ.
Howard (1966) models C as a uniform distribution on [0, 1], which implies
1
EðCÞ ¼ .
2
Next we relax what Howard did, and model the distribution of C such
1
that EðCÞ ¼ . We also follow Howard and model L as a uniform distribu-
2
tion on [0, 2].
1
We say that the base Howard example is L¼ U½0,2Š and EðCÞ ¼ .
2
1
The above gives us PðL > bÞ¼ ð2 bÞ, b 2 (0 for b > 2).
2
Of course, we do not consider b < 0 as discussed earlier. And so, we
arrive at:

1 1
,0 b 2 (9.8)
EðVjbÞ¼ ð2 bÞ b
2 2
5

2
1

We see that EðVjbÞ¼ b b +1 is a simple quadratic and that
2
2
d 5
4
db EðVjbÞ¼ b + ,so EðVjbÞ obtains a maximum of 9/32 when b ¼ 5/4
(Fig. 9.2).
We define:
b
dhVie ≜ max EðVjbÞ
b
From this definition, we see that when EðCÞ ¼ 0:5 and L¼ U½0,1Š:
dhVie ¼ 9=32
b
We are in agreement with everything that Howard has done to this
point. What we do not agree with is how he used the concept of clairvoy-
ance for additional information that may be learned. We note that the con-
cept of clairvoyance is also discussed in (Borgonovo, 2017, Chapter 11). We
return to this later in the chapter.