Database Reference
In-Depth Information
recommendations
k
is not too high in practice, mostly less than 10, so the inverse
F
1
can be determined robustly using, e.g., Newton's method, especially because
in the incremental methods, the preceding values of p can be used as starting values.
In practice, this gives rise to, e.g., the following modified method for determin-
ing the transition probabilities according to the adaptive approach (3.8) as we need
them in the ADP Algorithm 3.3.
Let us first consider the simple update of the single probabilities
j
p, where
j
is
the update step. This is displayed in Algorithm 4.1 (of course, there are many other
ways to calculate the transition probabilities).
Algorithm 4.1: Updating the single probabilities
Input: vector of single probabilities
j
p, index of the accepted recommendation
l
(
1 if none has been accepted), step size
α
j
Output: updated vector of single probabilities
j
+1
p
1: procedure UPDATE_P_SINGLE(
j
p,
l
,
α
j
)
2:
for
i ¼
1,
,
k
do
...
3:
if
i ¼ l
then
j
+1
p
i
:
¼
j
p
i
+
j
p
i
)
4:
α
j
(1
5:
else
j
+1
p
i
:
¼
j
p
i
+
j
p
i
)
6:
α
j
(0
7: end if
8: end for
9: return
j
+1
p
10: end procedure
For the probabilities of multiple recommendations, only the single probabilities
j
p are stored internally. After issuing the multiple recommendation, we first
compute the expected composite probabilities
j
p
¼
F
j
p
. Then, we update the
latter according to the accepted and rejected recommendations b
y v
irtue of Algo-
rithm 4.1, and we obtain the updated composite probabilities
jþ
1
p . The latter, in
, which is solved by means of the
turn, figure in the equation
jþ
1
p
¼
F
1
jþ
1
p
j
p as an initial guess, and we obtain the current single
Newton method with
probabilities
j
+1
p.
Algorithm 4.2: Updating the single probabilities from multiple
recommendations
j
p,
In
put:
vector
of
single
probabilities
issued
recommendations
a ¼ a
1
,
ð Þ
, index of the accepted recommendation
l
(
1 if none has
been accepted), step size
,
a
k
...
α
j
Output: updated vector of single probabilities
j
+1
p
(continued)
