In article <66b61316.0108091708.7d6b9958@posting.google.com>, S. F. Thomas <sfrthomas@yahoo.com> wrote:> robert@localhost.localdomain (Robert Dodier) wrote in message > news:<9kt895$rs$1@localhost.localdomain>... >> In the interest of brevity, I've indulged in wanton snippage, >> but I hope what's left yields something comprehensible.
>> S. F. Thomas <sfrthomas@yahoo.com> wrote:
>>> Robert Dodier wrote:
..............> Goodness, no. What I do argue however is that the semantics of > likelihood do not just fall neatly out from the semantics of > probability. Probability provides some of the underpinning, but not > all. Otherwise Fisher would not have been led up a blind alley by > asserting that the "likelihood of a or b is like the income of Peter > or Paul, we don't know what it is until we know which is meant."
I am by no means convinced that Fisher understood this, but I can see no way that the likelihood of "a or b" makes any sense at all. This> leads to a likelihood calculus in which set evaluation is of the form
> L( {a,b} ) = L(a OR b) = Max( L(a), L(b) )
Are you taking a view of a linear truth value system? AFAIK, this was first proposed by Lukasiewicz, and does not work at all well. Likelihood is NOT probability, and "a OR b" does not mean anything from the standpoint of likelihood.> which rather quickly proves to be inadequate. Had it not been > inadequate, I don't think classical statistics would have gone to all > the trouble it has to develop indirect methods of describing the > uncertainty in model parameters consequent upon sampling. Nor would > there have been a neo-Bayesian revival intended to supplant the > classicists precisely by offering a method of *direct* > characterization. Indeed, Bayes offers a likelihood calculus in which
> L(a OR b) ~ (L(a) + L(b))
Bayes never offered anything about a likelihood calculus. To Bayes, Fisher, Neyman, Laplace, Gauss, Kolmogorov, and others, one can take the or of statements or the union of events, but this is for probability. Likelihood is not probability, although it is an equivalence class of formal entities derived from probability.> where ~ is to indicate that some normalization, appropriate to the > construction of likelihood as a metaphorical (belief) probability, is > necessary. It is only with the fuzzy set theory that semantics > suggests itself
> L(a OR b) = L(a explains the data OR b explains the data)
> where "explains the data" is a fuzzy predicate no different in > principle from "is tall", and subject to calibration in conceptually > the same way. This leads, albeit with some reworking of the Zadehian > fuzzy set theory along the way, to
"Explains the data" is philosophical gobbledygook. Assuming that we can assume that we have a binomial model, and we get a positive number of successes and failures, ALL binomial distributions with 0 < p < 1 "explain" the data; there is a positive probability that the data could have come from such a model.> L(a OR b) = L(a) + L(b) - L(a)*L(b)
> where indeed the laws of probability are invoked, and at that in a > very simple way, but it is the fuzzy set semantics, and the device of > the calibrational proposition, that provides the essential frame that > Fisher overlooked.
The likelihood function can be multiplied by any constant, and often is; L and c*L are the "same" likelihood function for any statistical purpose. Why should anything be independent, even if it can be considered probabilities? ................. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558