From: ted.dunning@gmail.com   
      
   There are lots of ways to handle this kind of data.   
      
   One way is to convert your probabilistic data (with confidence factors)   
   into a deterministic data set by running through your original data   
   many times and producing new training examples that have a target   
   variable selected with the indicated probabilities.  This assumes that   
   your confidence factors are really probabilities which may be too big a   
   leap.   
      
   Another way to handle data with probabilistic training examples is to   
   note that most training algorithms are maximum likelihood estimators   
   that are maximizing the probability of the training data given the   
   model parameters (i.e. \hat \theta = argmax_\theta p(Y | \theta)).  For   
   independent training examples, this is just   
      
     \hat \theta = argmax_\theta log p(Y | theta)   
                       = argmax_\theta sum_i y_i log p(y_i | \theta) +   
   (1-y_i) log p(y_i | \theta)   
      
   Normally the target variables (y_i) are binary for supervised training   
   and because of this various other formulations that are equivalent to   
   the expression above are used to simplify later derivations.  If you   
   use this expression directly, however, then nothing says that the x_i   
   have to be binary.  In fact, there is a fairly well known theorem often   
   attributed to Gibbs that says that for two distributions p and q that   
   sum_i p_i log q_i has a maximum iff p = q.  But if p is the true   
   probability of something then sum_i p_i log q_i is just the expected   
   value of log q which explains why the maximum likelihood estimator   
   converges asymptotically to the actual probability distribution.   
      
   If this is a bit terse, please just say so.  The details are easy to   
   provide.   
      
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