Database Reference
In-Depth Information
transition pro
b
abilities of the single recommendations
p
ss
0
. This gives the transition
probability
p
ss
0
as
k
X
k
1
p
ss
0
¼ p
a
1
;
...
;a
k
ð
Þ
p
a
i
¼
ss
0
:
ð
4
:
3
Þ
ss
0
i¼
1
The advantage of this approach is that it is simple and linear with respect to the
single probabilities
p
a
i
ss
0
. However, for the case that we only work with probabilities
of state transitions associated with the recommendations, i.e.,
p
ss
a
, the presented
approach is not applicable as we will show later. We therefore introduce a more
sophisticated approach also based on Assumption 4.3.
4.2.2 Nonlinear Approach
For ease of reading, we will omit the product
s
from the indices and denote the
recommended or transition product by its index. Thus, we write
p
a
i
p
j
,
ss
j
¼
:
Þ
, and
p
a
1
,
...
,
a
k
p
1
;
...
;ð
j
.
In order to illustrate the problem, let us consider the case of two recommenda-
tions. For the case of the two single probabilities
p
1
:
¼ p
1
and
p
2
:
¼ p
2
,we
now need to determine the composite transition probabilities
p
1
:
¼ p
1
;ð
1
and
p
2
:
¼ p
1
;ð
2
. Without loss of generality, let us consider
p
1
, which we initially
determine as follows:
ð
Þ
a ¼
:
ð
1,
,
k
¼
:
...
ss
j
p
1
¼ p
1
¼ p
1
1
p
2
ð
Þ p
1
p
2
,
i.e., the probability of the transition for product 1 is the sum of the probabilities that
the user is interested in product 1 and not in product 2 and that he/she goes to
product 1, i.e.,
p
1
(1
p
2
), and that she is interested in both products and goes to
both of them, i.e.,
p
1
p
2
.
Now a user cannot, however, click on both recommendations at the same time;
so it is reasonable to model the second case as
p
1
p
2
p
1
p
1
þp
2
. The interest in the case
of both recommendations is thus multiplied by the probability that in this case, the
user decides in favor of product 1. This yields
p
1
p
1
þ p
2
:
p
1
¼ p
1
1
p
2
ð
Þ p
1
p
2
Similarly, we obtain for
p
2
p
2
p
1
þ p
2
:
p
2
¼
1
p
1
ð
Þp
2
þ p
1
p
2
