Database Reference
In-Depth Information
This description is somewhat simplified, as the delivery of the recommendation
a
has already been incorporated into the scaling via the coefficient
c(s, a)
; however,
the relationship of the values for the conditional probabilities
p
ss
0
in respect of
s
0
is
determined by each of the unconditional probabilities
p
ss
0
. We therefore assume that
there are some users who are not influenced by recommendations (because, for
instance, they are working from a shopping list). Although their overall influence is
already limited by the recommendation
a
(since there is another user group which
is open to the recommendations, hence the scaling factor
c
), their specific behavior
in the transition from
s
to
s
0
is unaffected.
It follows that instead of saving all transition probabilities
p
ss
0
, we only need to
save the conditional probabilities
p
ss
a
(i.e., between recommendation and
recommended product) together with the unconditional probabilities
p
ss
0
. Techni-
cally, this means that for every rule
s ! s
0
, both
p
a
s
0
ss
0
(i.e., the probability that the
recommendation of the product
s
0
be accepted) and
p
ss
0
(i.e., the probability that a
user goes from product
s
to product
s
0
without a recommendation
s
0
) are saved.
A similar method was proposed some time ago in [SHB05], but in a more
incomplete form. In particular, the coefficients
c(s, a)
were merely modeled as
c(s)
therein, which prevents adequate handling of down-selling, as we shall
shortly see.
the optimal recommendations. So as now to determine the complete transition
obtain
ð
X
s
0
6¼s
a
p
ss
a
þ cs
;
p
ss
0
¼
1
:
ð
5
:
4
Þ
Furthermore, from
X
p
ss
0
¼
1
s
0
follows the relation
X
p
ss
0
¼
1
p
ss
a
,
s
0
6¼s
a
which, when used in (
5.4
), finally enables the calculation of
c(s, a)
:
1
p
ss
a
1
p
ss
a
cs
ðÞ¼
;
,
ð
5
:
5
Þ
which represents the change of all transition probabilities induced by recommen-
dation
a
(except for the target state
s
a
).
