From: sebastianstern@wanadoo.nl   
      
   For a project of mine, I have recently become interested in Data Mining. One   
   common technique in Data Mining (used by on line book stores for example) is   
   the mining of Association Rules.  Each rule has the form   
      
     A => B   
      
   where A and B are sets of objects (e.g., books), and each rule can be   
   interpreted as stating "The possession of A implies the possession of B".   
   The 'degree of confidence' in a rule is defined as the conditional   
   probability that a subject (user) is interested in an objects B under the   
   condition that the subject already posesses objects A.  This confidence is   
   thus computed using the familiar formula for conditional probabilities:   
      
   confidence(A => B) :=  P(B | A) := P(A and B) / P(B)   
      
   Only those rules with a confidence above a certain threshold are then used   
   to recommend objects B to a subject that is already in the possession of   
   objects A.   
      
   So far so good.  However, the purpose of such a system is always to   
   recommend ever more and more objects, i.e., to increase an aggregate of   
   objects (the goal of an on line book store is to sell as much books as   
   possible, and buyers want to own more than one book).   
      
   Such a system does _not_ distinguish between _degrees_ of preference, i.e.,   
   it does not produce and _ordering_ of preference between different objects;   
   and this is the crux of my post.   
      
   (Note that if confidence(A => B) > confidence(A => C), this does not imply   
   that predicted_preference(B) > predicted_preference(C).  The ordering of   
   association rules (with the same condition) by confidence is not the same as   
   the ordering of objects by predicted preference.)   
      
   So, for my project I am looking for a way   
   (1) to infer or predict the prefencences of objects somehow, based on user   
   input (ideally, the subject would have to input as little data as possible   
   himself).   
   (2) and then order all objects by descending (predicted) preference, so that   
   (hopefully) the user would only have to look at the top one.   
      
   For input, the system could present two objects at a time and let the   
   subject choose which he prefers.  The choice of the subject would reflect   
   his relative preference for one of the two objects.  The preference relation   
   is a strict ordering relation between objects, parametrized on the subject   
   and time (but let us assume that the subject's preferences do not change   
   over time).   
      
     O1 <    O2   
         S,t   
      
   Alternatively, the input could consist of assigning a grade, or a monetary   
   amount to each object in some set of objects.  (This is actually just a way   
   of monotonically mapping the ordering relation between objects on the   
   ordering relation between numbers: value(O1) < value(O2) implies O1 < O2.)   
      
   Furthermore, if possible, I would like the subject to be able to input that   
   he prefers some object (a 'maximum object') above all possible objects, and   
   that he prefers some object below no object (i.e. the 'absent object').   
      
   So my questions are, roughly, these:  Has such a thing been done before?  If   
   so, could you provide me with some references to e.g. books and/or articles?   
   (I have looked into 'fuzzy association rules', but as far as I can tell   
   these do not meet my needs.  How should I input and represent the preference   
   relation?  What algorithms can be used to predict preferences?  Where can I   
   find out more?  Am I making any sense? ;-)   
      
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