How can I replace this condition by a probability?

Question

I want to see if a datapoint x should (or not) be assigned to a nearest component y using the following condition:

if ($d > T$) then {do not assign x to y}. With $d = distance(x,y)$ and $T = \bar{d}+\sigma$ where $\bar{d}$ is the mean distance of y to datapoints that were already assigned to it in the past, and $\sigma$ the associated stdev.

I want to replace this condition (i.e. ($d > T$)) by an equivalent expression which is expressed as a probability $p$. How can I do that ? I've tried for exemple $p = exp(\frac{-d^2}{2\sigma`^2}) / (\sigma`\sqrt{2\pi})$ but it does not behave like the first condition unless we manually find a good value for $\sigma`$

Note: it does not really matter if it is not a true probability (that sum to one).

Answer 1

If y is too far from x you should be able to define T independent of y. I wouldn't do it this way. But if you want to update sigma and T and then calculate P(d>T) you can and decide whether or not to keep y based on that probability.