Database Reference
In-Depth Information
¼
1
2
1
2
p
1
¼
0
p
1
þ p
1
p
1
¼
:
45
for the first recommendation and
¼
1
2
1
2
p
2
¼
0
p
2
þ p
2
p
2
¼
:
2
for the second recommendation. Both
pro
babilities are obviously lower than 50 %.
We now see the problem: if, e.g.,
p
1
has been estimated as 0.7, i.e., for both
recommendations
a
1
and
a
2
issued, the user in 70 % selects the first one, we would
obtain
p
1
¼
2
p
1
¼
1
4. This is obviously nonsensical. For this reason, the linear
approach can only by used meaningfully if we use all probabilities
p
j
instead of just
those associated with the recommendations
p
i
.
:
■
The entire expression can be extended without principal difficulty to
k
recommendations displayed together, even though the notation is somewhat
awkward:
X
Y
p
i
Y
j∈
k
\
Q
p
l
p
1
;
...
;k
ð
Þ
X
¼ F
l
p
1
;
...
;
ð
p
k
Þ ¼
1
p
j
ð
4
:
4
Þ
l
p
i
Q∈ fg[P
k
\
f
ðÞ
i∈Q
i
∈
Q
Here,
k
¼
f
1
;
...
;
k
g
is the index set of feasible recommendations, and P its
is the set of subsets of
k
that do not contain
l
united
power set, so
fg[ P
k
\
fg
with the singleton set of
l
.
In terms of our initial terminology, (
4.4
) may be expressed as follows:
,
¼ p
1
;
...
;k
ð
Þ
p
ss
0
¼ p
a
1
;
...
;a
k
ð
Þ
F
a
l
p
a
1
p
a
k
ss
a
k
s
0
¼ s
a
l
:
¼ F
l
p
1
;
...
;
ð
p
k
Þ ¼
:
ss
a
1
;
...
;
ð
4
:
5
Þ
ss
a
l
l
This implies that as opposed to the linear approach (
4.3
), the approach (
4.5
)
works out only for those successor products
s
a
l
that are associated with one of the
single recommendations
a
l
. A further disadvantage compared to the linear case is,
of course, the nonlinearity itself.
If we combine the probabilities as vectors p
¼
T
p
1
;
...
;k
ð
Þ
p
1
;
...
;k
ð
Þ
...
and
1
k
p
k
)
T
, this mapping of single into multiple recommendation probabil-
ities is expressed as a vector function F
¼
(
F
1
...
p
¼
(
p
1
...
F
k
)
T
p
¼
F
ðÞ:
ð
4
:
6
Þ
Since in practice, however, we can determine only p, we need the inverse
p
¼
F
1
ðÞ:
ð
4
:
7
Þ
Unfortunately, the function F from (
4.4
) is a nonlinear function. It is,
however, smooth and (component-wise) monotone. In addition, the number of
