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Prediction.
Let
x
1
=
x
1
...x
t
with
x
i
∈Q
,i
∈{
1
,...,t
}
be a context sequence
of questions and
x
t
+1
be the user's next question. Then, we define the
probability of observing question
x
t
+1
after
x
1
as:
∈Q
x
1
)=
P
(
˄
t
+1
|
˄
1
)
ˆ
1
)
˄
t
+1
,ˆ
t
+1
,x
1
)
,
P
(
x
t
+1
|
·
P
(
ˆ
t
+1
|
·
P
(
x
t
+1
|
(11)
where
˄
t
+1
=
p
˄
(
x
t
+1
) is the projection of question
x
t
+1
on the topic space
,
ˆ
t
+1
=
p
ˆ
(
x
t
+1
) is the projection of question
q
t
+1
on the space of learning objec-
tives
T
,
˄
1
=
p
˄
(
x
1
)=
p
˄
(
x
1
)
...p
˄
(
x
t
)and
ˆ
1
=
p
ˆ
(
x
1
)=
p
ˆ
(
x
1
)
...p
ˆ
(
x
t
),
x
1
being the last sub-sequence within topic
˄
.
L
2.3 Other Models
In order to show the effectiveness of the introduced learning-oriented question
recommender, we intend to perform a comparison with other models that use
Markov chains. One of the most widely used ones is a simple variable length
Markov chain (VLMC) over the question space, which we will further refer
to as Simple Recommender (SR). VLMCs were widely used in the literature
for prediction purposes in various application domains (e.g. data compression,
context-aware search, etc.) [
6
,
10
].
The Simple Recommender (SR)
is defined using a VLMC with random
variable
Q
over the question space
Q
(1:
t
)
)tobethetran-
sition model of the VLMC trained over a subset of the history database
. Consider
P
(
Q
(
t
+1)
Q
|
.In
order to learn a VLMC model, the algorithms presented in [6] were employed.
Then, for a given context of questions
x
1
=
x
1
x
2
···
H
x
t
with
x
i
∈Q
and a
, the probability of observing
x
t
+1
after
x
1
is given by:
new question
x
t
+1
∈Q
x
1
)=
P
(
Q
(
t
+1)=
x
t
+1
|
Q
(1)
=
x
1
,...,Q
(
t
)
=
x
t
)
P
(
x
t
+1
|
(12)
=
P
(
Q
(
t
+1)
=
x
t
+1
|
Q
(
tʻ
+1)
=
x
tʻ
+1
,...,Q
(
t
)
=
x
t
)
,
where
ʻ
=
ʻ
(
x
t
,x
t
1
,...
) is a function of the past determined during the learning
process of the VLMC. Let
D
= max
x
t
,x
t
1
,...
ʻ
(
x
t
,x
t
1
,...
) be the maximal mem-
ory length of the VLMC. Figure
5
represents a simple recommender model with
D
=3.
Fig. 5.
Simple question recommender based on a VLMC
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