Robert Dodier wrote:> > "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!
I ignored your "kewl" witticisms before; I do so again. But do continue to indulge yourself; there's always a place for jesters at court.> 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.
And?>> [...] 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.
Well... no. At the point at which an uninterpreted formalism is applied, semantics enters to either give meaning to the formula, or to make a nonsense of it. I continue to maintain that it is a semantic nonsense to have the random variable somehow supposed to vary conjointly with the model parameter, which in some sense is a summary of the entire distribution from which the random variable is supposed drawn. It takes a heroic, subjectivist stretch of metaphor to have it otherwise.> 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.
Well... no. It is a nonsense only up until some semantics is brought to it that gives it sense.>>> 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?
Under the probability calculus, you may do a change of variable from the random variable to some function transformation of it, in particular the loss function. Under the likelihood calculus, the same is possible, but the fact that likelihood is a point function, not a set function, renders the general rule for change of variable different -- easier in fact -- from what it is under the probability calculus. As to issues of risk and action, the notion of expected loss consequent upon any given action, is rendered as a possibility (or likelihood) distribution, or in effect a fuzzy set. In other words, after you average out the probabilistic chance elements by taking the expectation, the remaining, irreducible uncertainty, consistent with your prior and the data under the assumed model, is rendered in the form of a possibility distribution. Bayes would of course proceed to average out the possibilistic uncertainty as well. That though cannot a priori be justified, and it would certainly make a mess of your utility theory if the latter has no room for *fuzziness*, as opposed to chance. Anyhow, _Fuzziness and Probability_ goes into all of that, and in some detail. Btw, what> is a likelihood prior? It seems a charming contradiction in terms -- > some kind of mythical beast.
The idea is quite simple, and I've spelt it out somewhere else in this thread. It is that prior uncertainty is presumed based on some prior, informal experience with the phenomenon in question. Thus, for a Bernoulli process for example, a prior could be cast in principle as amounting to having observed b successes (more or less) out of m trials (more or less).>>> 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.
It is a figure of speech, Dodier. 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.
Of course. Bayes construes the probability *model* to be conditional pdf, with the model parameter construed as random variable varying conjointly with the random variable whose distribution it in some sense models.>>> 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.
Of course I did no such thing.>> 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.
What do you mean? And how does it relate to what we are discussing? (ii) The likelihood calculus which you state above looks> suspiciously similar to a rule derived from laws of probability.
It *is* derived from the laws of probability. You must have missed large parts of the thread while feigning sleep.>> 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.
What would have been your answer to Fisher when he said words to the effect: the likelihood of w1 or w2 is like the income of Peter or Paul, you can't know what it is until you know which is meant? 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.
First you claim boredom with the discussion, and say you're going to sleep, only to re-appear, apparently wide awake and engaged. Now you go off to work on "applications". You're most certainly welcome, but I would suggest you make a habit of saying what you mean, and meaning what you say.> Hmm, perhaps axiomatic contortions are like yoga: it seems strange until > you try it, then it's fun and relaxing.
Let me not keep you from your yoga then.> Regards, > Robert Dodier > -- > "Nature exists once only." -- Ernst Mach
Regards, S. F. Thomas