Database Reference
In-Depth Information
s
0
2S
a
¼ p
ss
ð
s
0
2S
a
. Similar from the transition index
l
,wederivetheindexes
l
c
¼
ss
0
s
0
∈S
a
and p
½
¼ p
a
s
0
p
½
ss
0
¼
u
l
,
s
l
∈
S
a
l
,
s
l
2 S
a
and
l
u
¼
.
1,
s
l
2S
a
1,
s
l
∈
S
a
Then we can reformulate Algorithm 5.3 in a more compact way (and computa-
tional cheaper) as Algorithm 5.4.
Thus, we first update the conditional probabilities of the recommended products.
Only in case that no recommendation has been accepted, the unconditional prob-
abilities of all non-recommended products are updated. Strictly speaking Step
4 requires additional scaling since the update of the unconditional probabilities of
the non-recommended products by Algorithm 4.1 may change their sum
X
s
0
2S
a
p
ss
0
.
However, this is an easy task. We just need to store the sum before and after the
update and then to multiply all updated probabilities by the corresponding factor.
Algorithm 5.4: Update of the internal from conditional probabilities for
multiple recommendations, Version 2
Input: vector of in
te
rnal probabilities
j
p
½
and fixed probabilities
j
Π
a
, delivered
recommendations
a ¼ a
1
,
ð
...
,
a
k
Þ
, index of product
tr
ansition
l
, step size
α
j
Output: updated vector of internal probabilitie
s
jþ
1
p
½
and
jþ
1
Π
a
1: proce
du
re UPDATE_P_DP_MULTI2(
j
p
½
,
j
Π
a
,
a
,
l
,
α
j
)
jþ
1
p
½
2
:
¼
UPDATE_P_MULTI
c
(
j
p
½
,
a
,
l
c
,
α
j
)
⊳
update of recommendation probabilities
c
3:
if
l
c
¼
1 then
jþ
1
p
½
u
:
¼
UPDATE_P_SINGLE
(
j
p
½
u
,
l
u
,
4:
α
j
)
⊳
update of non-recomm. probabilities
5:
else
jþ
1
p
½
u
j
p
½
u
6:
:¼
⊳
unchanged take-over of non-recomm. probabilities
7:
end if
jþ
1
j
8:
Π
a
:¼
Π
a
⊳
unchanged take-over of the fixed component
return (
jþ
1
p
½
,
jþ
1
9:
Π
a
)
10:
end procedure
Although Algorithms 5.3 and 5.4 deliver in principal the same result, they
represent different perspectives on the update of conditional and unconditional
probabilities. While Algorithm 5.3 updates both probability types by one algorithm,
Algorithm 5.4 separates this calculation and allows calculating conditional and
unconditional probabilities by different update algorithms. Even in case of the same
update rule (
3.8
), as used in Algorithm 5.4, this may result in different counters for
the coefficients
α
j
and lead to different results. We will not further deepen this
special topic here.
