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In-Depth Information
Fig. 5.2 Example of a
product with two
recommendations
a ¼ a
1
; að Þ
and a total
of five following products
s
a
1
a
1
s
'∈
S
a
a
2
s
a
2
a
3
s
s
a
3
a
4
a
5
s
'∉
S
a
s
a
4
s
a
5
and we obtain the updated conditional probabilities
jþ
1
p
a
. To the latter, we apply the
inverse mapping
jþ
1
p
½
¼
F
1
j
p
ssa
and obtain the current internal probabilities
jþ
1
p
[
a
]
. Subsequently, we carry over
j
p
ss
a
unchanged to the next update.
jþ
1
p
a
Algorithm 5.1: Update of the internal from conditional probabilities for one
recommendation
Input: vector of internal probabilities
j
p
[
a
]
and fixed probability
j
p
ss
a
, delivered
recommendation
a
, index of product transition
l
, step size
α
j
Output: updated vector of internal probabilities
jþ
1
p
[
a
]
and
jþ
1
p
ss
a
1: procedure UPDATE_P_DP_SINGLE(
j
p
½
,
j
p
ss
a
,
a
,
l
,
α
j
)
j
p
a
:
¼
F
j
p
ss
a
(
j
p
[
a
]
)
2:
⊳
conversion into conditional probabilities
jþ
1
p
a
:
¼
UPDATE_P_
SINGLE(
j
p
a
,
l
,
3:
α
j
)
⊳
update of conditional probabilities
⊳
jþ
1
p
½
:¼
F
j
p
ss
a
1
jþ
1
p
4:
conversion into internal probabilities
jþ
1
p
ss
a
j
p
ss
a
5:
:¼
⊳
unchanged take-over of the fixed
component
return (
jþ
1
p
[
a
]
,
jþ
1
p
ss
a
)
6:
7:
end procedure
5.2.3.2 Multiple Recommendations
We shall now attend to the case of multiple recommendations. Let again
a ¼ a
1
,
ð Þ
be the
k
issued recommendations and
S
a
be the set of states
corresponding to the former; see Fig.
5.2
. Let further
S
a
¼ S
AðÞ
=
...
,
a
k
S
a
be the comple-
mentary set of all not-recommended successor states, i.e.,
s
0
2S
a
. We denote the fixed
