A Fuzzy Probabilistic Method for Medical Diagnosis

Abstract

The max-min composition in fuzzy set theory has attained reasonable success in medical diagnosis in the past thirty years for estimating the probability of a patient diagnosed with a certain disease. However, there has been no theoretical justification why the method would work. We create a theoretical model to calculate the probabilities of hypothetical patients having designated diseases, and use simulated dataset to explain why the max-min composition has been successful. In addition, based on the theoretical model, we propose a fuzzy probabilistic method to estimate the probability of a patient having a certain disease. The proposed method may produce a more accurate estimate than the max-min composition.

Appendix

Physicians sometimes base their decisions on exclusion of certain diseases given the present symptoms [20, 21]

Using the max-min composition, the chance that a patient does not have a certain disease would be given by [20]

$$ {\mu}_{\mathrm{T}}\left(\mathrm{p},\ \mathrm{d}'\right)= \max \hbox{-} \min\ \left[{\mu}_{\mathrm{Q}}\left(\mathrm{p},\ \mathrm{s}\right),\ \left(1 - {\mu}_{\mathrm{R}}\left(\mathrm{s},\ \mathrm{d}\right)\right)\right] $$

(A1)

In our model, the probability of a patient not having disease d, P _p (d’), will be calculated as

$$ {\mathrm{P}}_{\mathrm{p}}\left(\mathrm{d}'\right)={\sum}_{\mathrm{s}}\left[{\mu}_{\mathrm{Q}}\left(\mathrm{p},\ \mathrm{s}\right).\ \left(1-{\mu}_{\mathrm{R}}\left(\mathrm{s},\ \mathrm{d}\right)\right)\right] $$

(A2)