Database Reference
In-Depth Information
we have formally introduced the vector function F
G
:
W ! V
:
p
G
¼
F
G
p
ð :
ð
4
:
8
Þ
In the same way, from (
4.7
), we derive the inverse F
G
:
V
F
V ! W
:
p
G
¼
F
G
p
ð :
ð
4
:
9
Þ
Example 4.2
To make the discussion less abstract, we give a very simple example.
Consider the case where we have only two products, i.e.,
n ¼
2. Then we have
two single
r
ecommen
da
tions
a
1
and
a
2
and three multiple recommendations
a
1
¼ að
,
a
2
¼ að
,
a
3
¼ a
1
;
ð
a
2
Þ
. This yields the following probability spaces:
n
n
o
0
p
a
i
o
,
p
a
1
p
a
2
p
a
3
p
a
3
ss
2
ss
0
1,
p
a
3
ss
1
þ p
a
3
V
:
¼
ss
1
;
ss
2
;
ss
1
;
ss
2
1
0
p
a
s
0
:
p
a
s
1
p
a
s
2
W
:
¼
;
ss
0
1
ss
1
ss
2
Let us consider the first “multiple” recommendation
a
1
¼ að
. Then the relation
p
a
1
p
a
1
¼
F
a
1
simply translates into
¼
F
a
1
¼
:
p
a
1
ss
1
p
a
1
ss
1
p
a
1
ss
1
For the global function F
a
G
, we get
¼
¼
F
a
G
:
p
a
1
ss
1
p
a
2
ss
2
p
a
1
ss
1
p
a
1
ss
1
Same holds for the second “multiple” recommendation
a
2
¼ að
. For
t
he thi
rd
(a
nd
only real) multiple recommendation
a
3
¼ a
1
;
Þ
, the mapping p
a
3
¼
F
a
3
ð
a
2
p
a
1
now becomes more complex:
0
1
p
a
1
ss
1
p
a
1
ss
1
1
p
a
2
ss
2
þ p
a
1
ss
1
p
a
2
!
¼
F
a
3
¼
@
A
ss
2
p
a
1
ss
1
þ p
a
2
p
a
3
ss
1
p
a
3
ss
2
p
a
1
ss
1
p
a
2
ss
2
ss
2
:
p
a
2
ss
2
p
a
2
ss
2
1
p
a
1
ss
1
þ p
a
1
ss
1
p
a
2
p
a
1
ss
1
þ p
a
2
ss
2
ss
2
In this case, F
a
3
is also our global function, i.e., F
a
G
¼
F
a
3
. Thus, we arrive at
the complete global mapping:
