Database Reference
In-Depth Information
of measuring success and control groups is addressed in Chap.
11
). Since no
recommendations are issued here, Algorithm 4.1 may straightforwardly be used
to determine the
p
ss
0
.
Control groups, however, are not always available. Furthermore, the approach is
prone to inconsistencies. Hence, we shall jointly estimate both the conditional as
well as the unconditional probabilities invoking our central Assumption 5.2.
5.2.3.1 One Recommendation
We shall first consider the case of one recommendation
a
issued in state
s
and
introduce the following notation for the internally used probabilities:
p
ss
0
,
s
0
6¼ s
a
p
ss
0
,
s
0
¼ s
a
p
½
ss
0
¼
:
For a fix
p
ss
a
Assumption 5.2 with (
5.3
) included stipulates the following 1-1
mapping between our internal probabilities and the conditional ones:
8
<
1
p
ss
a
1
p
ss
a
ss
0
p
ss
0
,
s
0
6¼ s
a
ð p
ss
0
,
s
0
cs
;
6¼ s
a
p
½
p
ss
0
¼ F
p
ssa
¼
¼
:
ð
5
:
10
Þ
p
ss
0
,
s
0
¼ s
a
:
p
ss
0
,
s
0
¼ s
a
(The special cases
p
ss
a
¼
1 and
p
ss
a
¼
1will not be addressed here, since they are
cumbersome to deal with and do not figure decisively in the basic approach.)
Introducing the vectors p
[
a
]
[
a
]
)
s
0
, p
a
¼
(
p
ss
0
)
s
0
and letting
m
be the number
of successor states
s
0
,(
5.10
) defines the vector function F
p
ssa
¼
¼
(
p
ss
0
T
F
p
ss
a
...
F
p
ss
a
:
:
p
a
p
½
¼
F
p
ssa
ð
5
:
11
Þ
Since in practice, however, we can update only p
a
, we also need the inverse
p
½
¼
F
1
p
ssa
p
a
ð :
ð
5
:
12
Þ
which is equally easy to compute. Equipped with this mapping, we proceed in
essentially the same fashion as Algorithm 4.1 from Sect.
4.2
.
We store only the internal probabilities
j
p
[
a
]
, where
j
is again the update step, in
our rule base. After the recommendation
a
has been issued, we first compute the
expected conditional probabilities
j
p
a
. The unconditional probability
of the issued recommendation is kept fix, since we cannot update it anyway, as the
recommendation has been issued! The conditional probabilities
j
p
a
are now com-
puted by virtue of Algorithm 4.1 according to accepted or rejected recommendation,
¼
F
j
p
ss
a
j
p
½
