sfrthomas@yahoo.com (S. F. Thomas) wrote:> "Wise" <weiz@spam-block.sympac.com.au> wrote:
>> [...] So can a "fuzzy expert" panel of 100 vote on a particular elements >> inclusion in a fuzzy set (ie that a height is tall) and the extent to >> which they agree (say 70%) on the inclusion of that element in the >> set be converted to a membership value ? >> >> Then IF the answer is yes why then is this not probabilistic ? > > Asked and answered, as they say in court. Yes it is probabilistic, but > it doesn't make the membership function a probability distribution. It > makes the membership function what I have called a semantic likelihood > function, a point function that does not integrate to unity as > required of a probability distribution, but does range on [0,1] as > required of a membership function.
If we follow through on this notion of semantic likelihood, we'll find that the laws of probability already contain the appropriate rules for combining fuzzy statements. I'm comfortable with that, but it does make fuzzy logic less of a breakthrough.> I don't think people have appreciated the full implications of that > insight just yet. It certainly is *not* to say that fuzzy is just > probability after all, anymore than it makes sense to say that > likelihood *is* probability. (The Bayesians, in effect, do the latter, > but they are wrong IMO.) [...]
If we take that first step of making membership a likelihood function, we're already done. The laws of probability specify how likelihoods should be combined with each other, so there's no need to go off and invent a new logic for that purpose. Incidentally, probability also gives the rules for finishing the job: how to combine likelihoods with probability distributions to obtain a posterior over any variable of interest, linguistic or otherwise; the posterior over non-linguistic variables is what is needed to compute risks and take actions, assuming that losses are stated in terms of what's actually happening as opposed to what somebody says is happening. BTW Bayesians don't say that likelihood functions are probability distributions, so I don't know where the strawman argument above comes from. I'm actually quite sympathetic to the notion of linguistic uncertainty, calibration, making inferences from one vague statement to another, etc. It's just that I don't see any good reason to invent some adhockeries to treat the problem; probability works just fine. Take a look at my treatment of "Where in the world is Elvis Presley?" in Chapter 1 of my dissertation: http://civil.colorado.edu/~dodier/papers/dodier.ps.gz See also the discussion of sensor models, Chapter 6, for some more models of the "I can't quite get at what I really want to see" flavor. Hey, I'm just doing my part to wipe out insomnia. :) Regards, Robert Dodier -- "Nature exists once only." -- Ernst Mach