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: >>> [...] 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 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. > > Suppose, then, that I have a possibility or likelihood for two > different actions. Can I say that one action is preferable to > the other? If so, how do I determine which is more preferable?
Sometimes it is very clear which is preferable, sometimes less so. If you are minimizing loss then the smaller the centroid of the possibility set, the more preferred; however, the centroid is an insufficient measure of preference, for the largeness of spread also comes into the picture, with smaller spread (i.e lesser fuzziness) being in general preferable to larger spread. In a single-criterion, single decision-maker (DM) problem, this can go to the heart of the insufficiency of the Bayesian paradigm, and indeed explain why a (rational) decision-maker may choose neither to take, nor place, the Bayesian bets. Suppose for two actions, the corresponding possibility sets on expected utility have the same centroid, but one has greater fuzziness than the other, then the rational thing to do is to opt for the action with lesser fuzziness, no? Contemplating a Bayesian betting scenario, a decision-maker always has as an option the certainty of status-quo, i.e to neither take the bet nor place the bet, either of which options would presumably carry some residual possibilistic uncertainty deriving precisely from the modeling uncertainty which is sought to be illuminated by the Bayesian analysis. That is the single-criterion, single DM problem. In the more usual case, the optimal action in any given situation must be evaluated on more than one criterion. And quite often in practical decision-making, we have multiple decision-makers, and various ways of attempting to resolve differences among them. The utility calculus will take one nowhere fast in attempting to address these questions. With the likelihood/possibility calculus, it is in fact possible to address these questions, as inter-personal comparisons, both of belief and of preference, are far easier to address within such a framework. _Fuzziness and Probability_, in the part of it that elaborates an approach to decision analysis under uncertainty, attempts to do just that.>>>> [...] 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? > > In the world of models implicit in your statement above, there are > sampling distributions for observable variables and there are > parameters that govern those distributions. Some models are that > simple, yes. There are many models which don't fit into this neat > division of labor. Does every class of models require its own > reasoning calculus?
At the very least, there is deductive reasoning, and associated logical calculus, and there is inductive reasoning, and associated logical calculus. These two suffice in my view to carry the burden of any discourse concerning any object phenomenon, or system of inter-acting phenomena, that may be of interest. However, there is a third kind of reasoning which must remain outside either of these. It is the reasoning that derives from *insight* and which leads us in a very mysterious fashion to posit the intension models and associated premises/hypotheses that may then be subject to various kinds of deductive and inductive massaging. There is also a fourth, which is what we are here engaged in, which is a kind of meta-reasoning.>> (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. > > Well, I have no problem with deriving fuzzy reasoning from probability, > but I thought that was precisely what you were arguing against.
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." 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) ) 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)) 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 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. Btw, there is a school of fuzzy which is the mirror-image of what it is you seem to wish to maintain. They maintain that fuzzy is logically prior to "crisp". This misses an essential point in my view. And that is that a bivalent logic is perfectly capable of generating ever-higher levels of fuzziness, in exactly the same way that you can run fuzzy models on binary computers. The ultimate bivalence of the computer does not disable it when it comes to elaborating computable fuzzy models; in fact the reverse. The important distinction is not between bivalent logic and multivalent logic, but between meta-language and object language. A bivalent logic in the meta-language is perfectly adequate for the purpose of modeling the fuzziness in the object language. The essential notions of membership function, and of rules of combination (fuzzy union, intersection, etc.) all belong to the bivalent meta-language we must recall, and all our theorems are cast in the bivalent meta-language. And to even suggest casting it in a logically primitive multivalent meta-language would be hopelessly confusing in my view. Instead, from the vantage point of a bivalent meta-language, it is possible to see, at the level of the object language, a class of crisp terms which are very clearly a special case of fuzzy. But that is in the object language. And that observation does not render fuzzy prior to crisp; rather it is the crispness of the meta-language that permits us to bootstrap our way to the higher reaches of fuzziness in successions of object languages. I make this point because probability stands in a similar relation to fuzzy, and to likelihood. Just because we use probability to generate the membership or likelihood function, it does not follow that there is no value-added in making the leap from the one to the other. And I maintain that there is an essential duality between the two, with the distinctness, yet connectedness, that that implies.>> [...] First you claim boredom with the discussion, and say you're >> going to sleep, only to re-appear, apparently wide awake and engaged. > > I always feel like a million bucks after a good nap.
Well, in that case, Dodier, I hope the present offering again succeeds in putting you gently to sleep.> Regards, > Robert Dodier
Regards, S. F. Thomas