One of the most common questions regarding fuzzy set classification is how does it relate to statistical classification. What is the difference between the degree of membership (also known as possibility) in the set and the probability of being in that set? Once again refer to the fuzzy FAQ.
A good example that demonstrates the conceptual difference between statistical and fuzzy classification is the one given by Bezdek in the reference mentioned previously. In the example, a person who is dying of thirst in the desert is given two bottles of fluid. One bottle’s label says that it has a 0.9 membership in the class of fluids known as non-poisonous drinking water. The other bottle’s label states that it has a 90% probability of being pure drinking water and a 10% probability of being poison. Which bottle would you choose?
In the example, the "probability bottle" contains poison. This is quite plausible since there was a 1 in 10 chance of it being poisonous. The "fuzzy bottle" contains swamp water. This also makes sense since swamp water would have a 0.9 membership in the class of non-poisonous fluids. The point is that probability involves crisp set theory and does not allow for an element to be a partial member in a class. Probability is an indicator of the frequency or likelihood that an element is in a class. Fuzzy set theory deals with the similarity of an element to a class.
Both are valid approaches to the classification problem. If we were to classify someone as "old", fuzzy membership makes much more sense than probability. If we were to classify the outcome of a coin flip, probability makes much more sense.
Both fuzzy systems and neural networks attempt to determine the transfer function between a feature space and a given class. (Note: if the reader is unfamiliar with neural networks, an excellent overview by Dr. Leslie Smith can be found here.) Both can be automatically adapted by the computer in an attempt to optimize their classification performance.
One difference between the two methods is that the membership functions of a fuzzy classifier can be initialized in a state close to the correct solution. What this means is that a fuzzy classifier can be set up by a skilled HCI designer to do a pretty good job of classification even before the classifier is adjusted by the computer. A neural network, however, can only be initialized in a random state. Thus, the training of the computer to optimize the classifier is usually much faster with a fuzzy classifier than a neural network classifier.
The problem with a fuzzy system is it is difficult to deal with too many features, membership functions, and/or rules. Neural networks, are highly suited for large amounts of features and classes.