Reasons not to use fuzzy numbers instead of pds to represent uncertainty

Question

Can someone explain why (if at all) it would be a bad idea to use fuzzy numbers in order to represent uncertainties in model parameters instead of probability distributions?

To motivate my question - assume a decision model where the parameters are provided as probability distributions. The information to support the decision is computed by performing a Monte Carlo simulation that evaluates the model for each sample drawn from the parameter distributions and gives the result in terms of the computed sampled probability distribution. From the resulting pd, one can compute the expected value, risk, and other statistical measures.

Could one simply use fuzzy numbers in place of pds and apply fuzzy arithmetic instead of MC?

Answer 1

Well, fuzzy numbers, and fuzzy logic in general, is mean to represent partial membership or partial truth (as opposed to classical logic where things are or are not true)..in fuzzy logic, the objects themselves are vaguesly defined whereas with probability, we are representing incomplete knowledge of a well defined object (e.g., the mean).

The mathematics of each are different as well. See http://en.wikipedia.org/wiki/Possibility_theory. For example, the probabilty distribution of X + Y given PDFs will be quite different from the possibility or fuzzy set function of X + Y when they are represented by possibility functions.

I guess a bigger quesiton is why do you want to use fuzzy logic vs probability? Are you concerned with error rates or classification?

The nice thing about probabilities is that they have the ability to be calibrated to real-world impacts. For example, if your resulting distriubtion has expected value = 6, then, if your model were correct, you should expect that the average result in the real world will tend toward 6. I don't know if Fuzzy logic can be related to actual observations so readily...my impression is that it is more of decisionmaking tool along the lines of game-theoretic type formulas. I could be wrong though.