Reconstructing Distances Among Objects From Their Discriminability
We describe a principled way of imposing a metric representing dissimilarities on any discrete set of stimuli (symbols, handwritings, consumer products, X-ray films, DNA strings, etc.), given the probabilities with which they are discriminated from each other by a perceiving system, such as an organism, person, group of experts, neuronal structure, or technical device. As it turns out, to reconstruct the metric of dissimilarities one does not have to assume any of the constraints underlying the traditional methods, such as multidimensional scaling or cluster analysis. In particular, one does not have to assume that discrimination probabilities are monotonically related to distances. Nor does one have to assume that the discrimination probabilities are symmetric, or that the probability of discriminating a stimulus from itself is a constant. The only requirement that has to be satisfied is regular minimality, a principle we propose as the defining property of any discrimination process: for ordered stimulus pairs (a,b) , b is least frequently discriminated from a if and only if a is least frequently discriminated from b. This is joint work with E. N. Dzhafarov (Purdue University).