Database Reference
In-Depth Information
probabilities
p
ðaÞ
ss
00
of a very new product
s
00
may deteriorate our whole approach
(
5.8
). To overcome this problem we will present some first ideas here.
At this, we replace
n
p
ðaÞ
ss
0
with the “stabilized” probabilities
n
p
ðÞ
ss
0
,
if n n
min
p
ðÞ
ss
0
n
e
:¼
,
ð
5
:
22
Þ
0,
if n
<
n
min
where
n
min
is a threshold value for the minimum statistical mass (usually 20 or
more) and instead of (
5.8
) now calculate
X
s
0
6¼s
a
e
1
e
p
ss
a
1
e
q
π
s
p
ss
a
r
ss
a
þ
e
ðÞ¼e
;
p
ss
0
r
ss
0
:
ð
5
:
23
Þ
p
ss
a
There remains a problem at (
5.23
) however because of the fact that for new
transitions the conditional action value
p
ss
a
r
ss
a
initially is 0 or small. That means that
its recommendations are scarcely delivered and
p
ss
a
r
ss
a
can scarcely grow (unless
e
q
π
e
ð
increases via its unconditional action value). We then have a vicious circle.
In order to escape this, we modify (
5.22
) for the conditional probabilities
p
ss
0
as follows:
s
;
n
p
ss
0
,
if n n
min
n
p
ss
0
:¼
,
ð
5
:
24
Þ
s
C
m
p
ss
0
,
if n
<
n
min
where
s
C
∈
) is a fixed scaling factor. Here the
n
refers to the counter of the
conditional probability (i.e., for delivery of
a
), whereas on the other hand
m
is the
counter for the unconditional probability! In this way we replace (
5.23
) with
the final estimation
[1,
∞
X
s
0
6¼s
a
p
ss
0
r
ss
0
:
p
ss
a
1
e
1
^
q
π
s
ðÞ¼p
ss
a
r
ss
a
þ
;
ð
5
:
25
Þ
p
ss
a
We now come to the interpretation of (
5.25
). Since the unconditional probabil-
ities
p
ss
0
are continually updated, even without delivery of
s
0
, these have real
chances of being delivered as recommendations, and the conditional probability
counter increases. As soon as it reaches the threshold
n
min
, the initial auxiliary
probability
p
ss
a
is replaced by the conditional probability
p
ss
0
.
The scaling factor
s
C
should be motivated in the broader sense. If
s
C
¼
1 is set,
there is the risk that the new recommendations are often not sufficiently strong to be
shown. (Generally
p
ss
a
>
p
ss
a
is the case, i.e., the probability of the transition to a
product
s
a
is generally higher if it is also recommended.) It follows from this that in
general
s
C
>
1 should be selected, so that the transition probabilities
p
ss
a
have a real
chance of being delivered.
