I've been studying a particular problem recently, and it seems like there might be techniques from statistics or ML that could be applied. Any advice or comments would be appreciated.
We're given some set of values sampled from a particular distribution. The values themselves are hidden to us, but their frequencies are known. We don't know the exact distribution the hidden values are drawn from, but using other information we've developed a model/approximation of it that probably has some errors, both in terms of missing elements and the frequency of a particular element. Our eventual goal is to map elements of our approximate distribution to their hidden counterparts in our target distribution.
For some small number of hidden values, we're also given the actual member of the distribution this value corresponds to. In a sense, a few of the hidden things are 'revealed' to us, but most of them remain hidden.
Is there any way for us to take the information we get from the revelation of the small number of values from our 'real' distribution and use it to refine our approximation of the entire target distribution, and not just the values that were revealed?
Obviously we know with certainty which elements of our approximate distribution correspond to the hidden values that were revealed, but can we use the fact that we also know 'how far off' our approximation was for those values to make our approximation more accurate for other values?