Is there a theory of probability based on fuzzy set theory?

[Ray12]

One thing I've wondered for quite some time, and looked for, but not found anything I consider adequate is whether there is a theory of probability based on Zadeh's fuzzy set theory.

The closest I've found is Zadeh's Perception Probability Theory, but this doesn't quite seem to cut it.

Does anyone here know of any theories that successfully generalize probability theory from a boolean set theory foundation to a fuzzy set theory foundation?

 

[SW VandeCarr]

One thing I've wondered for quite some time, and looked for, but not found anything I consider adequate is whether there is a theory of probability based on Zadeh's fuzzy set theory.

The closest I've found is Zadeh's Perception Probability Theory, but this doesn't quite seem to cut it.

Does anyone here know of any theories that successfully generalize probability theory from a boolean set theory foundation to a fuzzy set theory foundation?

Fuzzy sets are defined as a pair $(A,m)$ where $m:A\rightarrow[0.1]$ so it would seem that it could be a basis for some kind of probability theory (PT). But formally a random variable is defined as a function which maps a real number from the interval [0.1] to an event space. I'm not familiar with FST enough to know if the common distributions functions used in PT applications (including Bayesian statistics) have any application in FST. For example, how is a sigma algebra defined on a set in FST?. From what little I read on this, FST is based on "perception" rather than measurement.

http://kmh-lanl.hansonhub.com/uncertainty/meetings/zadeh03vgr.pdf

I'm not aware of any generally accepted alternative PT that is not based on the Kolmogorov axioms. I do understand that fuzzy and many valued logic{s} have been successfully employed in industry.

 

[Stephen Tashi]

One thing I've wondered for quite some time, and looked for, but not found anything I consider adequate is whether there is a theory of probability based on Zadeh's fuzzy set theory.

One of the strengths of fuzzy set theory is that it is not a probability theory. An element of a fuzzy set that has a 0.7 degree of membership in it, does not have a 0.7 probability of being in the set and a 0.3 probability of not being in the set. It has a definite degree of membership equal to 0.7. So, for example, a house might have a 0.7 degree of membership in the set of "colonial style houses". If you treat the degree of membership as something determined by a random variable, the you can ask questions like "what is the probability that a randomly selected house has a degree of membership in the set of "colonial style houses" that is between 0.7 and 0.75? Such things are handled by ordinary probability theory.

So it isn't clear what one would mean by "a theory of probability based on Zadeh's fuzzy set theory" because fuzzy set membership does not resemble probability and ordinary probability theory can be applied to fuzzy set membership.

Did you have specific goals or ideas for a generalization of probability theory that somehow incorporates fuzzy sets in a new way? It would be interesting to discuss.

 

[Ray12]

I do know that fuzzy set theory and probability aren't the same thing, just FYI (note: I don't mean for this to sound rude or anything, limitations of the written word and all that.)

Regular Probability theory is based upon boolean (two valued) set theory. As far as I am aware of, this limits the questions that probability can accurately answer to those that are precisely defined (such as what's the probability that it will rain or not today).

However, it appears to me that if one were able to generalize probability theory to one based on fuzzy set theory, which can handle imprecision well, then the probability that would result would also be able to handle vagueness.

For example, if I were to ask the question "what is the probability that it will rain today?" Where rain is understood to be imprecisely defined and a matter of degree (light drizzle, moderate rain, heavy downpour, etc.) Boolean based probability would, to the best of my knowledge, be unable to adequately handle this.

As far as I know, the only way that a boolean set based probability could give answers to this question that would approximate reality, would be to add further boolean sets (such as very light drizzle, light drizzle, ...)
However, this approach is disadvantageous in that in order to approximate the actual situation with boolean sets/events it would require a large number of sets, with a resultant increase in complexity.

It seems to me, that it would be more efficient to utilise fuzzy sets/events, as in this regard fewer sets would be needed to handle the imprecision and give a reasonably accurate answer.

 

[Stephen Tashi]

Ordinary probability theory applied to a continuous random variable representing the "degree of rainy-nes of a day" couild represent a greater variety of days than "rainy" and "not rainy".

Ordinary probability can deal with qualities like temperature, height, inches of rain. It's an interesting question whether Fuzzy Set theory really has any content that is not already captured by continuous variables that define such qualities. I think what makes Fuzzy Set theory a specialized method of talking about such qualities is that it has definite rules for the degree of membership of unions and intersections of fuzzy sets.

Fuzzy set membership is boolean in the sense that a statement like "Today has a 0.7 degree of membership in the set of rainy days" is a statement that is either true or false and not a little bit of both.

 

[Ray12]

Fuzzy set membership is boolean in the sense that a statement like "Today has a 0.7 degree of membership in the set of rainy days" is a statement that is either true or false and not a little bit of both.

I'm not denying that certain statements are or can be made crisp, such as the example you gave of: "Today has a 0.7 degree of membership in the set of rainy days" That is a case where I would agree that, yes, it is clear cut true or false.

I'm just wondering about the cases where things have a large amount of imprecision and (probably due to practical constraints) would be difficult to render extremely precise (in the lack of vagueness sense).

And where I see things really getting interesting is when you have both Fuzzy AND Probabilistic uncertainty. (Note: I do consider these two, at least semantically, distinct.)

Again, to give an example, consider a situation where one was standing at a door to a building watching people walk through. And one were to wonder, "What is the probability that the next person to enter will be bald?"

In this case you have fuzzy uncertainty as a result of the word bald being imprecisely defined (and if one were to precisely define it, and box it in a boolean set, you would run the risk of a Sorite's paradox.)

You also have probabilistic uncertainty in this example, as you can't be completely sure or not whether the next person to enter will be bald(ish) or not (without cheating and peeking outside or something).

 

[Ray12]

Ordinary probability theory applied to a continuous random variable representing the "degree of rainy-nes of a day" couild represent a greater variety of days than "rainy" and "not rainy".

.

Also, I am unfamiliar with exactly how this would work. Could you explain how it would adequately detail an imprecise situation, or direct me to any good papers or articles on the subject? You've got my curiosity piqued.

 

[SW VandeCarr]

You also have probabilistic uncertainty in this example, as you can't be completely sure or not whether the next person to enter will be bald(ish) or not (without cheating and peeking outside or something).

Interesting. I can't say I have a real understanding of FST, but this helps to clarify things. PT will simply define a person as bald or not bald and assign a probability P(b). Everyone is either b or ~b where P(~b)=1 - P(b). FST will classify every man into one of several ordered categories such as full, mostly full, half bald, mostly bald, bald. The putative degrees of membership in the "bald set" would be 0.00, 0.25, 0.50, 0.75, 1.00. Would this be correct? I must say, if this is all there is to it, PT is quite capable of modeling this given some consistent method of classification.

One difference I see is that in FST each degree is precise in that there is no uncertainty or "variance" associated with the classification. The classification may be wrong, but that's not issue for the theory. If meteorologists estimate 15 cm of snow "plus or minus" 3 cm, they are not "wrong" if 18 cm actually falls. Likewise, if they predict an overnight low temperature of 4C and the actual low is 2C I don't think they can be said to wrong even without stated error bars. I'm not sure how FST would handle weather forecasts if it relies on perception rather than measurement.

EDIT:

There's a fair amount of literature on analyzing subjective assessments. Here's an example:

http://www.ncbi.nlm.nih.gov/pubmed/2024718

 

[Stephen Tashi]

Could you explain how it would adequately detail an imprecise situation, or direct me to any good papers or articles on the subject?

What's the definition of adequate detailing? Any time you have a probability distribution over a continuous random variable, you have about as much detail as can be provided for a situation that is imprecise, in the sense of being indeterminate. I'll look for papers where people have applied probability theory to the degree of membership in fuzzy sets when I get time.

I tend to think about fuzzy set theory in the practical context of process control. We begin with a description of the control actions in ordinary language, such as "If the room is somewhat hot, turn on the fan to medium and pull the shades down a little". The degrees of membership for things like "the shades" in a set like "down-ness" are usually not specified numerically and the problem becomes to use empirical or theoretical data to assign numbers to these degrees of membership and to associate these degrees of membership with quantities that can be measured (such as the length to which the shades are pulled down, what "medium" is in RPM, etc.). So the control problem is a big parameter-fitting task where the degrees fo membership are included in unknown parameters.

About all that fuzzy set theory does for you in this process is to specify functional relations among the degrees of membership in intersections, unions, and complements of the fuzzy sets. For example, if you have natural language phrase like "if the room is hot and stuffy", there is an implied intersection between the sets of "hotness" and "stuffy-ness". If you assign membership x in "hotness" and membership y in "stuffy-iness", the membership in "hotness and stuffness" is determined as min(x,y). Sometimes people use other functions that min(x,y) for intersections. Sometimes people enforce additional rules that relate statements in natural language to functions of membership. For example, one could stipulate that if the the degree of membership for a "somewhat hot" room in the set of "hotness" is x then the degree of membership for a "just a little hot" room in the set must be x^2.

In the practical context of fuzzy control, I can't think of any situation involving indeterminancy that is not adquately handled by ordinary probability theory. For example, if the degree of membership of the room in the set of hot rooms is a probabilistic event, then we can assume it is modeled by some probability distribution on the interval [0,1]. In your example, if you assume that the "degree of baldness" of the man is given by a probability distribution on [0,1], does that fail to capture something about the situation?