"S. F. Thomas" <sf.thomas@verizon.net> wrote, in part:> Robert Dodier wrote: >> 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. <snip> > > [...] By identifying likelihood with the membership function of a fuzzy > set, it is at last possible to suggest different likelihood semantics, > [...] we are led to a different rule for evaluation of the set hypothesis > {w1, w2}, the rule of product-sum, specifically L(w1)+L(w2)-L(w1)*L(w2).
Acolytes, make way. Arise, ye laity. O mystic formula, make thyself known! That's funny -- it looks just like the formula one derives by assuming that memberships are likelihoods and likelihoods are defined in terms of conditional probabilities.> [...] Bayes theorem is properly applied to random variables that vary > conjointly at the same level of semantic indentation, for example height > and weight of the population of adult males. The model parameter does not > vary conjointly with the random variable whose distribution it in some > sense models. [...] The parameter and the random variable whose > distribution it models are at different levels of semantic > indentation.
Like any other mathematical result, Bayes' theorem is properly applied to anything that fits, in a formal sense: the variables in the theorem are just placeholders. In another time, you would have been one of those people who asserted that the square root function is properly applied only to positive numbers.>> 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. > > The extended likelihood calculus can achieve the same kind of result, > with with or without (likelihood) priors.
OK, now this is something I haven't heard about -- how does the extended likelihood calculus take loss, risk, and action into account? Btw, what is a likelihood prior? It seems a charming contradiction in terms -- some kind of mythical beast.>> BTW Bayesians don't say that likelihood functions are probability >> distributions, so I don't know where the strawman argument above >> comes from. > > Bayes construes the likelihood to be conditional pdf.
I dare say Bayes does no such thing; he's gone on to his eternal reward, you know. Let me correct the slight misunderstanding of how a likelihood function is defined. It goes like this: Consider a conditional probability p(A|B). Let f(b) = p(A=a|B=b). Then we say f is a likelihood function. A likelihood function is not guaranteed to integrate to unity, so it can't be a probability distribution, conditional or otherwise. This much is well known to Bayesians and non-Bayesians alike.>> 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. > > Of course probability works just fine.
Thanks for conceding my point. I appreciate it.> But probability (over sample space) gives rise to likelihood (over > parameter space) and the calculi required to manipulate the two are > different.
(i) This betrays a very limited view of what a model can be: apparently there are but samples and parameters. Many interesting models are not so simple. (ii) The likelihood calculus which you state above looks suspiciously similar to a rule derived from laws of probability.> And the semantics of likelihood has always been problematic. WHich is > why [...] Bayesians go through all the axiomatic contortions they do to > treat likelihood as though it were probability.
Speak for yourself. I don't have any problems with the semantics of likelihood functions. Now you're free to have problems with anything you like; meanwhile, I think I'll go off and work on applications, as described in the aforementioned dissertation. Hmm, perhaps axiomatic contortions are like yoga: it seems strange until you try it, then it's fun and relaxing. Regards, Robert Dodier -- "Nature exists once only." -- Ernst Mach