Collaborative filtering using a linear model Consider I have a set of movies and a set of users ($A$,$B$,$C$,$D$) and a matrix with related scores (I can have gaps in this matrix).
Consider a linear regression model where a specific user A's rating is meant to be a weighted average of other users ratings:
$$R_A = a + w_B R_B + w_C R_C+ w_D R_D$$
where $R_A$ is a rating of user A and so on, $w_B$ is the weight parameter for rating $R_B$ and $a$ is a normalization factor.
The following is the table I have got: 
Usersm1m2m3m4m5m6m7m8    A 3 4 5   1      B  5 5 5    1    C  2   5 5 3 4    D 4 5 4 4    2
Question: How can I derive a parameter estimation method and use it to obtain the four weights $w_B$, $w_C$, $w_D$?
 A: I think this is not getting answers because it looks like you're trying to address a well-known problem using an approach that doesn't seem very appropriate.
I see the "self-study" tag so it may be that you're just learning about the techniques... so that's the angle I'm taking when answering.
It looks like you're ignoring the fact that this is a recommender system, and just addressing the problem under the hypothesis: A's ratings can be predicted by a linear combination of the ratings by users B,C, and D.
In this case you just have a standard linear regression problem with 4 datapoints (A's existing ratings). Now I have a feeling that a regression of 4 parameters from 4 datapoints will not give you very good results... just not enough data.... and what you'll get as a result will not be useful to predict the rest of the ratings.
In a recommender system where ratings by one user are predicted from ratings by other users on the same item (movie, etc.), you're doing (a basic form of) user-user collaborative filtering, and the standard approach is that you predict user A's ratings from the top-k most similar other users. 
Therefore, your weights are either uniform (1/k) (you take the mean of those ratings) (or 0 if the user is not among the top-k most similar), or dependent on the similarity (ratings from user Z are weighted according to the similarity between A and Z, calculated from other movies they've both rated).
You can take a look at this paper on collaborative filtering, it may help.
