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can be simulated from the histogram
of
U
's and
V'
s rating differences Hist(
R
UJ
-
R
VJ
) for
From this equation, we can see that error
e
I(
V
). Thus, Hist
(
R
UJ
-
R
VJ
) serves as the correlation measure between
U
and
V
. For rating ranges
from one to five, Hist(
R
UJ
-
R
VJ
) is a distribution of nine values, i.e. from
8
J
2
I(
U
)
\
4to4.
Compared with similarity measures, it preserves more details in friends' review
ratings. Compared with a joint distribution approach, it has fewer degrees of
freedom.
Assuming
U's
and
V's
rating difference on the target item
I
is consistent with Hist
(
R
UJ
-
R
VJ
). Therefore, when
R
VI
has a rating
r
VI
on the target item, the probability
that
R
UI
has a value
k
is proportional to Hist(
k
r
VI
).
Pr
R
UI
¼
ð
k
j
R
VI
¼
r
VI
Þ1
Hist
k
ð
r
VI
Þ
:
(4.9)
When the target user
U
has more than one immediate friend who co-rates the
target item, the influences from all of those friends can be incorporated in a product
of normalized histograms of individual friend pairs.
Z
Y
V
1
1
Z
V
Hist
Pr
R
UI
¼
ð
k
j
f
R
VI
¼
r
VI
:
8
V
2
U
ð
I
Þ\
N
ð
U
Þ
g
Þ ¼
ð
k
r
VI
Þ
;
(4.10)
where
Z
V
is the normalizing constant for the histogram of each immediate friend
pair, and
Z
is the normalizing constant for the overall product.
Once we
A
0
¼
a
0
I
obtain Pr
R
U
¼
ð
k
j
Þ
;
Pr
R
I
¼
ð
k
j
A
¼
a
u
Þ
;
and Pr
R
UI
¼
ð
k
j
f
, these probabilities are fed into an aggregator
where the ultimate rating distribution of
R
UI
is shown in (4.3).
R
0
UI
, the predicted
value of
R
UI
, is the expected value of the distribution.
R
VI
¼
r
VI
:
8
V
2
U
ð
I
Þ\
N
ð
U
Þ
gÞ
X
R
0
UI
¼
A
0
¼
a
I I
;
k
Pr
R
UI
¼
k
j
A
¼
a
U
;
f
R
VI
¼
r
VI
:
8
V
2
U
ð
I
Þ\
N
ð
U
Þ
g
K
(4.11)
4.4.3 Distant Friend Inference
We have just introduced the approach for predicting a target user's rating of a target
item from those of the user's immediate friends for the same item. However, in
reality, there are many cases where no immediate friends of a target user have
reviewed the same target item; thus, the rating of the target user cannot be predicted
from immediate friend inference. To solve this problem, we propose distant friend
inference.
The idea of distant friend inference is intuitive. Even though
V
, an immediate
friend of a target user
U
, has no rating for the target item, if
V
has his/her own
immediate friends who rated the target item, we should be able to predict
V
's rating
of the target item via the immediate friend inference, and then to predict
U
's rating
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