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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?

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  • $\begingroup$ You know a histogram of the data then? It is confusing as stated. $\endgroup$
    – vonbrand
    Mar 7 '16 at 3:19
  • $\begingroup$ Yeah, we have a histogram of both the hidden values and the values in our model. $\endgroup$
    – pg1989
    Mar 7 '16 at 4:24
  • $\begingroup$ So essentially you want to learn the inverse cumulative distribution function (or quantile function)? That way you can estimate $x_i = F^{-1}(p_i)$? $\endgroup$ Mar 7 '16 at 19:40
  • $\begingroup$ Exactly, but we also want to revise our model based on the observed difference between our model and the revealed values. $\endgroup$
    – pg1989
    Mar 7 '16 at 20:48
  • $\begingroup$ @pg1989, are you able to sample from your approximate distribution? $\endgroup$ Mar 7 '16 at 23:04
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I would suggest using bipartite matching. Have one vertex on the left side for each hidden value you observed. Have one vertex on the right side for each value in your approximate distribution (your model for the underlying distribution). Draw an edge between each pair of vertices $(u,v)$, with a weight based on how similar their counts (frequencies) are. Look for a matching of maximal weight.

This will work if you have no other information about the hidden values, other than their frequencies. If you also have some information about their order or other side information, I don't know how to use bipartite matching for that.

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