Define a posterior probability of y given x when the model is not probabilistic

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Suppose we have a very simple online k-means where each new data-point is assigned to its nearest center (the mean is updated incrementally). Each center (cluster) is labelled with the most common label of data-points assigned to that cluster. In this special configuration: is it possible to compute a sort of "posterior probability"? I.e., can the posterior probability of a class label $y$ given a data-point $x$ ($P(y|x)$) just be $1/\text{distance}(x, m_y)$, where $m_y$ is a center labelled with $y$ which is nearest to $x$?

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Try and explain your question to someone who might have some interest, but doesn't already understand everything that you know.   Also, maybe this would be better on stats.stackexchange.com – Anthony Quas Nov 8 at 23:56
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