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I have a task to make a decision, say to classify an object as $X$ or $ \overline X$. However, $\overline X$ usually means everything else and you only have positive examples of $X$ and not so many negative examples of $\overline X$. So estimation of probability distribution of $\overline X$ is hard. But the good thing is that the space has the metric defined on it.

Common sense suggests that something non-probabilistic might make much more sense there, in particular since metric is defined, for example SVM is much more suitable for this task because you can estimate the margin to reject $\overline X$. However, there is no complete discussion on that.

I'm looking for a book or survey paper to discuss this problem and non-probabilistic approaches to it compared to probabilistic ones.

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  • $\begingroup$ Added a note about metric defined on the space, this seems to be the key point. $\endgroup$
    – Nikolay Shmyrev
    Commented Apr 20, 2016 at 11:09
  • $\begingroup$ Added some papers to on non-probabilistic theory. $\endgroup$
    – user75138
    Commented Apr 26, 2016 at 15:08

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Generally, decision theory has been dominated by probabilistic approaches. However, as explained in this wiki article, there are alternatives out there. The most notable are fuzzy logic and possibility theory.

Some relevant papers you will want to read through: https://arxiv.org/ftp/arxiv/papers/1301/1301.2271.pdf

http://dl.acm.org/citation.cfm?id=979604

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