Database Reference
In-Depth Information
set
of
unconditional
probabilities
assigned
to
the
recommendations
as
Π
a
¼ p
ss
f
s
0
∈S
a
.
Firstly, we again introduce our internal probabilities:
p
ss
0
,
s
0
2S
a
p
½
ss
0
¼
,
p
a
s
0
ss
0
,
s
0
∈
S
a
i.e.,
p
½
ss
0
corresponds to
p
½a
ss
0
, where the grouping is now over a multitude of issued
recommendations. For the special case of one recommendation
S
a
¼
sfg
, the two
coincide.
The question, which also figures in the simulations described in Sect.
5.4
,
is
:
How can we calculate the transition probabilities of multiple recommendations
p
ss
0
from those of single recommendations
p
ss
a
a
nd
p
ss
0
?
(4.3). Alternatively, we may resort to the nonlinear approach (4.5).
Linear Approach
In conjunction with the linear approach (4.3), we generalize (
5.10
) and again obtain
the 1-1 mapping between the internal and conditional probabilities:
8
<
:
k
X
k
1
s
0
cs
að p
ss
0
,
;
2S
a
¼
¼
k
X
k
1
i¼
1
0
1
p
½
ss
0
p
a½
ss
0
p
ss
0
¼ F
Π
a
F
s
a
i
X
k
1
k
i¼
1
p
a
s
0
@
A
,
s
0
ss
0
þ
cs
að p
ss
0
;
∈
S
a
i¼
1
,
a
i
6¼a
s
0
8
<
1
p
a
i
ss
a
i
1
p
ss
a
i
k
X
k
1
s
0
p
ss
0
,
2S
a
i¼
1
0
@
1
A
,
s
0
¼
:
ð
5
:
13
Þ
1
p
a
i
ss
a
i
1
p
ss
a
i
:
X
k
1
k
p
a
s
0
ss
0
þ
p
ss
0
∈
S
a
i¼
1
,
a
i
6¼a
s
0
As for the one-recommendation case, we change over to the vectors p
½
¼
p
½
ss
0
T
¼ p
ss
0
s
0
and introduce the vector function F
Π
a
¼
and p
a
F
1
F
m
Π
a
Π
a
...
s
0
through
p
a
¼
F
Π
a
p
½
ð
5
:
14
Þ
