FUZZINESS AND PROBABILITY
The book "Fuzziness and Probability" by S. F. Thomas is now in print.
This is a revised version of a manuscript that was made available
by internet email in December, 1994. The internet version has been
superseded and is therefore no longer made available.
1995; 320 pages; ISBN 0-8050-2356-0; soft cover; $29.95
Publisher: ACG Press, PO Box 782948, Wichita KS 67278-2948, USA
Tel:316-777-4425
Fax:316-689-6889
Email: acg@acginc.com
OVERVIEW
I claim in the book:
o to have found the extended likelihood calculus that eluded
Fisher, and generations of statisticians since
o to have corrected the mistake in the Zadehian fundamentals of
fuzzy set theory which put him at odds with Aristotle -- the
laws of excluded middle and contradiction are restored,
without undoing the essential fuzziness of natural language
descriptors
o to have corrected the mistake in the Bayesian inference
theory that treated universals (uncertainty about models) and
particulars (uncertainty about the chance of occurrence of
individual events) symmetrically, while keeping the essential
idea that sources of evidence -- prior belief plus
experimental data -- may be combined to give a posterior
expression of belief, scientific or otherwise, about the
universals involved
o to have found the solution to the nagging foundational
problem of measurement theory -- how to address errors of
measurement within the theory itself. First you change the
paradigm from one of assumed precision of measurement to one
where imprecision (fuzziness) is the general case. It is
easy for precision to fall out of a paradigm of imprecision,
but quite difficult to make imprecision fall out from within
a paradigm of precision.
There are other claims besides, but these are the main ones,
and the last one is key: we are dealing here with a paradigm
change. At the foundations of science, where we are addressing
issues of measurement -- including the measurement of attributes
of probability, belief, utility -- and issues of scientific
inference, where again we must confront issues of probability and
its measurement, the paradigm in place is one of precision, the
idea that the total ordering axiom applies, that sample spaces may
be continuous, that points within sample spaces may be discerned
with absolute levels of precision, and that they may be measured
in principle to an infinite number of decimal places. Once this
notion is relaxed, everything changes. One then finds oneself at
a point where everything in science seems to converge (or from
where everything diverges) and simultaneously in the realm of
inference throry, probability theory, measurement theory, fuzzy
set theory, decision theory, even the foundations of deductive
logic, of the philosophy of truth, of semantics, and so forth. In
much the same way that Einstein's theory of relativity changed the
analytic paradigm for the theory of motion by putting the observer
into the picture, theories of inference regarding real-world
phenomena change when the imprecision inherent in the use of
language for the conveying of measurement reports -- the language-
use phenomenon itself -- is explicitly brought into the picture.
And, as with Einstein's theory of relativity, which did not
diminish the usefulness of the Newtonian theory in most everyday
applications, the change of paradigm here explored will not change
the usefulness of the paradigm of precision in those application
areas where "adequate" precision is capable of being achieved for
the attributes of concern. Equally, however, the change of
paradigm appears necessary in areas where the attributes of concern
(eg., subjective probability, utility) are not susceptible of
sufficiently precise measurement/description.
While the breadth of coverage seems immodest to say the
least, that's where I was led, purely serendipitously, as I have
indicated in the Preface and Introduction. It's all due to Zadeh,
as the idea of fuzziness started it all, and is at the center of
the whole thing. But surprisingly, the connectedness of all these
things seems to have eluded Zadeh, who in insisting on the
distinction between probability and fuzziness seems to have cut
himself off from exploring the intimate relation between the two.
I think I have exposed the connection, which far from weakening
the power and beauty of Zadeh's master stroke, strengthens it even
further.
EXCERPTS
>From the Introduction:
------------
"What I offer the reader, in sum, is a philosophical essay setting
out what insights I think I have encountered in addressing the
problems associated with the scaling of judgmental attributes for
use in the analysis of decision problems. In addressing these
problems, I have found it interesting to explore foundational
issues in the fuzzy set theory of semantics, the theory of
statistical inference, the theory of measurement, along with the
concepts of probability, likelihood, possibility, logic,
combination of evidence, and other topics typically of interest to
those concerned with exploring the foundations of science. The essay
is primarily addressed to academicians and practitioners of the
'decision sciences' -- broadly construed to include statistics,
economics, management science, operations research, and like
disciplines -- who are interested in exploring foundational issues
arising in the area of nexus between probability and fuzziness.
It should also be of interest to social scientists, in particular
those who address issues related to the psychometric scaling of
judgmental attributes, and to engineers and researchers in the
areas of artificial intelligence and cybernetic systems.
The plan of the essay is as follows: Chapter I provides some
motivational background, giving a brief review of the controversy
in the foundations of statistical inference, showing the way in
which fuzzy sets may enter the picture, while pointing out that
at least one difficulty encountered by the likelihood school of
statistical inference -- the paradoxes sometimes engendered by
marginalization by maximization -- may apply equally to the fuzzy
set theory, where this method of marginalization is commonly applied.
Chapter II puts forward a discussion of the philosophical
fundamentals underlying the theory -- of semantics,
measurement, phenomena, models, and probability. Chapter III
develops a non-Zadehian fuzzy-set theory of semantics, in which,
among others, the laws of excluded middle and contradiction
are restored. Chapter IV is an elaboration of the possibility
calculus in its application to inductive (statistical) inference.
Chapter V continues the elaboration of the possibility calculus,
as applied to deductive inference. Chapter VI addresses the
problem of decision analysis under uncertainty, and shows how the
possibility calculus may be applied. Chapter VII provides a
comparison of the possibilistic inference and decision theory
with the Bayesian. Finally, the essay is concluded in Chapter
VIII with an attempt to put the whole development in perspective."
>From Chapter I ("Motivational Background"):
------------------------------------
"This essay develops three basic propositions: first, that data
are fuzzy sets in general, not point observations; second, that
uncertainty about probability models or model parameters is
structurally similar to semantic uncertainty about fuzzy data or
fuzzy descriptions; and third, that uncertainty in data and
uncertainty in modeling both yield to a possibility calculus which
provides a common interpretive framework within which to construct
a theory of inference and apply it to problems of decision
analysis under uncertainty.
The essay therefore brings together two diverse streams of
literature, one dealing with statistical inference, the other with
fuzzy sets and approximate reasoning. Out of the confluence of
these two streams flows a theory of semantics and possibilistic
inference, freed from anomalies present in both streams. Their
joining should therefore be to the enrichment of both."
>From Chapter II ("Philosophical Fundamentals")
-----------------------------------------
On semantics:
"It should be clear that the view of semantics being developed
here implicitly distinguishes _calibrational_ propositions from
propositions in _actual use_ -- what I call later, for lack of a
better term, _descriptional_ propositions. In the foregoing
discussion, presenting users of the language with exemplars of
various height values u , and asking whether they would use the
descriptor 'tall' to describe the height value u is a calibrational
exercise. When the descriptor 'tall' is actually used, as in the
eyewitness's description 'the perpetrator is tall', we have a
_descriptional_ proposition, in actual use. If the population
usage of the term 'tall' has been calibrated, we have the basis for
constructing a 'possibility distribution' or 'semantic likelihood'
function for the unknown value of the perpetrator's height.
The distinction between calibrational and descriptional propositions
is thus the distinction, as with measuring instruments, between
calibration, and use."
On fundamental measurement:
"The crux of the problem lies in the treatment of error. The
problem of equivalence intransitivity stems from the intrinsic
accuracy or sensitivity limitations of the judge or measurement
device -- what may be called intrinsic error. Errors of
observation (Topping, 1955) may include as well accidental and
systematic elements. The classical formulation of the problem of
measurement runs into difficulty because it does not allow, within
the formulation itself, for the treatment of error. Intrinsic
error is not allowed because it is inconsistent with the
fundamental ordering axiom which is usually assumed -- a practical
necessity within a paradigm of point representation. Systematic
error is not allowed because the formulation deals in principle
with primary measurement whereas systematic error is a problem
only for secondary measurement devices defined by reference to a
primary standard -- more a practical problem than one of
principle. Accidental error is treated, not as part of the problem
of measurement in principle, but as an extra-theoretical problem
of statistics. Once again, the premise of point measurement is too
rigid an ideal to allow variations of measurement to be considered
within the theoretical framework; variations must be explained
through the invocation of random or accidental occurrences, when
the real culprit is more usually intrinsic accuracy limitations.
Within the fuzzy set framework being proposed for the problem of
measurement, it is possible to treat all three forms of error.
Systematic error and intrinsic error would be revealed by means of
the kind of calibration exercise already discussed ... What would
be regarded as accidental error in the classical framework is
treated not so much as 'error', but as variations in measurement
reports wholly consistent with the intrinsic imprecision inherent
in those reports." ......
"The point of all this is quite simply the contention that the
character of an attribute space is determined fundamentally by an
act of abstraction that precedes subsequent description
(measurement) of objects with respect to the attribute, or
comparison of objects according to the degree of possession of the
attribute. This effectively achieves a separation between the
issues of representation and uniqueness, on the one hand, and the
issues pertaining to the empirical idiosyncracies of a particular
judge or measurement device, on the other hand. The representation
and uniqueness issues are now purely abstract, and therefore
clear-cut (although not necessarily simple) while problems of
error and equivalence intransitivity which reside essentially in
the perceptional domain may be dealt with by the use of fuzzy
descriptors, which allow for imprecision -- hence error and
occasional intransitivity -- in judgment."
On probability:
"More generally, I would claim that the criterion of repeatability
as a way of dividing the respective areas of application of the
frequency view and the axiomatic view does not have the validity
that it would seem to have on the surface. In the first place, no
experiment may literally be repeated, since things change (time,
at a minimum) from one experiment to the next. If we make another
throw of a pair of dice, or pick another card from a pack, or grow
another crop on an experimental plot, we have not repeated an
earlier trial; at the very least, time has intervened, and at
worst, some determinant of the response variable may have changed,
accounting for the changed result. Yet it is meaningful to talk of
a repetition, for there is something that stays the same from one
trial to the next. That something is what I would call the
'morphology' we mentally construct around the phenomenon. In
typical frequentist statistical experiments, such a morphology is
quite explicit -- the population of occurrences from which
observations are drawn is well defined, and the variables of
interest, response variables and explanatory variables are well
defined. In the case of the question concerning the European war,
nothing is defined -- neither the population of occurrences of
which the European war may be deemed to be an instance, nor the
precise variables that we should look at to help us make a
judgment and answer the question posed. Nevertheless, when we
engage in discourse concerning the phenomenon, we implicitly
construct such a morphology. One may offer some such discussion as
follows: in any war, it is the relatively stronger of the
combatants that usually wins, and the greater the relative
strength of one combatant over the other, the faster that
combatant wins. The strength of a combatant depends upon the
number and quality of men, weapons, and war materiel that it
possesses, and the cohesiveness and morale of the troops, as well
as on the strategic deployment of such forces. And so on. It is
not my intention here to develop a theory of war or combat, only
to illustrate my point that in the elaboration of discourse
concerning a phenomenon, we implicitly construct a mental
morphology around it, defining a population of 'objects' --
combatants, considered in pairs -- and a set of attributes or
variables which enter either as response variables of interest --
relative strength in this case -- or as explanatory variables --
number and quality of men, etc. Once we have such a morphology, we
have the basis for a frequentist approach to probabilistic
modeling of the phenomenon, for it allows us to speak in general
terms about events that would otherwise be considered simply
unique. Thus, on the one hand, the repeatability of unarguably
repeatable statistical experiments really derive from the
morphology of the situation, and on the other hand, once we do
construct a morphology for seemingly unique events, they also
become individual instances, or repetitions, of occurrences that
fit into a larger pattern. I conclude that morphology is the key
concept that allows us to close the gap between the frequency view
and the axiomatic view in the application of the probability
concept."
"Once we have a morphology, the next question is on what basis do
we assign probabilities to events within the morphology: is it
objective or subjective? This categorization is irrelevant to the
concept of probability that I have in mind. A probability model is
in general purely a hypothesis which we are prepared to entertain
as having some descriptive power in summarizing the relative
frequencies of occurrences in the population of real or realizable
occurrences defined by the morphology with which we structure the
phenomenon and order our observations. As a hypothesis, it hardly
matters how a probability model is arrived at, whether by
subjective introspection, or by an empirical goodness-of-fit
procedure on an experimentally observed sample from the population
of interest. Whenever formal experimental observations are not
possible I see no harm in making use of subjective introspection
(of one's accumulated experience), but the status of such
'subjective probabilities' in the present concept of probability
is of the subjective estimation of a frequency probability. The
probability being subjectively estimated is not assumed to have a
psychological origin, as it would if it were taken as a primitive
concept.
I disagree, further, with the axiomatization of subjective
probability which proceeds from the assumption that subjective
probability judgments can be totally ordered, and that therefore
sharp, numerical judgments of probability can always be elicited
from any human judge. As discussed in the previous section, I
would distinguish the abstract attribute space associated with the
attribute of probability, which I have no difficulty conceiving of
as a totally ordered linear continuum, from the empirical
idiosyncracies and sensitivity limitations of the judge making the
probability judgments. That is to say probability judgments may
be fuzzy, although the underlying abstract concept of probability
allows probabilities to be totally ordered numerical points
precise to an infinite number of decimal places. Furthermore, even
in the straightforward situation where we need not rely on
subjective probability judgments, but instead experimental sample
data are available, probability estimates may nevertheless remain
essentially fuzzy. As mentioned previously in Chap. I, the
likelihood function is essentially a fuzzy set describing what the
data say about the 'true' unknown probability model which may be
under investigation."
>From Chapter III ("Fuzzy Set Theory of Semantics"):
---------------------------------------------
"Zadeh (1975) has taken the position that the notion of grade of
membership is merely a subjective estimation of the extent to
which any given element may be said to belong to any fuzzy set in
question. On this view it is difficult to decide whether the set
of tall men, for example, could not simply by represented as a
Bayesian subjective probability distribution over the space of
height values. It is also difficult to establish any particular
set of combination rules. The minimum-maximum rules of the
Zadehian calculus have intuitive appeal, but lead to debatable
consequences. In particular, the self-contradiction law, the law
of contradiction, and the law of excluded middle are violated.
Should they be, and if not, what other rules of combination may we
substitute that would restore these laws. Zadeh has also taken
the position that the concepts of probability and fuzziness are
distinct, raising the question what kind of statistical methods
could one logically apply to establish either a membership
function, or any particular rules of combination which one might
care to propose.
As I have indicated previously, I find it difficult to accept the
idea that membership functions may be entirely subjective. If I
were to assert that 6ft. is short for a man, I think one would be
entitled to question whether I were a competent speaker of the
English language. Thus there is an element of convention in the
meaning of words in a language. As I have tried to point out in
the previous chapter, if a convention exists, then one's
subjective estimation cannot be the whole story -- there must be
an external reality out there regarding language use susceptible
of objective characterization.
Like Watanabe (1978), I also find it difficult to accept the
result of the Zadehian min-max calculus that the fuzzy term 'tall
and not tall' should be anything less than the logical absurdity,
as the law of contradiction requires. One would lose all
credibility as a witness in court if one were to testify that the
burglar was 'tall, but not tall'. The fuzziness of the term 'tall'
is not sufficient, in my view, to persuade a jury that such a
contradiction could have positive meaning within the English
language convention. Similarly, the law of excluded middle
requires that the disjunction 'tall or not tall' should be the
constant tautology. Again, I am not persuaded that the fuzziness
of the term tall is sufficient to justify the result of the min-
max calculus under which not all elements of the universe have
full membership, tautologically, in this disjunction. Finally,
the law of self-contradiction requires that any term which implies
its own negation must be the logical absurdity. Under the min-max
calculus, any term, not necessarily the absurdity, whose
membership function is everywhere less than half must imply its
own negation.
In what follows I depart from the Zadehian fuzzy set theory first
at the philosophical level. I take the membership function to
represent a usage convention which may in principle be objectively
determined, using statistical methods. Proceeding from this basic
assumption, the concepts of probability and fuzziness may indeed
be distinguished, but the concept of fuzziness is derived from
that of probability, in almost exactly the same fashion that the
Fisherian concept of likelihood derives from probability. And in
the same way that the concepts of probability and likelihood are
distinct, the concepts of probability and fuzziness are distinct,
though related concepts. Coincidentally, it turns out that
proceeding from such an assumption makes the min-max calculus
quite simply untenable as a set of universal rules. It becomes
clear that other rules are sometimes appropriate, and it forces
one to address the issue when does which apply. In so doing, the
law of contradiction and the law of excluded middle are upheld as
a matter of necessity, a happy result if one is disposed to accept
these as having positive empirical significance as semantic law.
The law of self-contradiction does not hold, as in the Zadehian
calculus, if one takes the containment relation between fuzzy sets
as representing the implication relation. If, however, as in fact
is necessary in the present development, the rule of implication
is defined with reference to possibility distributions rather than
membership functions, and possibility distributions, like
likelihood functions are unique only up to similarity
transformations, then the law of self-contradiction may be
restored.
This result, and the restoration of the laws of contradiction and
of excluded middle, are happy byproducts, however, rather than
starting objectives. My basic goal is rather to harmonize the
essential truth of the fuzzy set concept with the essential truth
of the concept of probability, and to try to sort out the
respective limits of application of the two concepts in the
representation of uncertainty.
The approach is axiomatic -- material axiomatic rather than formal
axiomatic: primitive concepts are first introduced and explained,
followed by empirical postulates, followed by lemmas and theorems.
In contrast with a formal axiomatic development, where primitive
terms remain uninterpreted, the primitive concepts and empirical
postulates introduced in the following development are intended
very much to have positive, empirical significance where semantics
is concerned. We try to be faithful to our own conceptualization
of phenomena and models, and give a morphology for the language-
use phenomenon followed by an extension-set model embodied in the
postulates which we adduce.
In pursuing this axiomatic approach, we necessarily accept the
rules of the two-valued set theory in our mathematical
metalanguage, while developing rules which we hope apply to the
fuzzy, many-valued terms which populate our object language. This
is a matter which itself requires some reflection. I defer this
discussion (see later, Sect. 3.4.5)."
On harmony between bivalent metalanguage and fuzzy object
language:
"It is by no means obvious that vagueness and fuzziness in natural
language should fall ultimately under the ambit of a two-valued
logic. Giles (1971, p. 322) for example has written that the
notion of the fuzzy set is prior to that of set. In the same
vein, Goguen (1974, p. 514) writes: 'Ideally we would like a
foundation for fuzzy sets which is independent of ordinary set-
theory ...' Goguen proceeds to axiomatize fuzzy set theory in the
language of category theory, but as a theory of semantics it
remains non-empirical. Within the empirical framework which is
proposed in this development, it would appear that what is fuzzy
with respect to U , e.g. 'tall', can be rendered as something
crisp in
U
[0,1]
e.g. mu[TALL]: U -> [0,1].
Evidently we could define fuzzy sets of type 2, as Zadeh has done,
corresponding to fuzzy sets with fuzzy membership values, in which
case what is fuzzy with respect to
U
[0,1]
could be rendered as something crisp in
U
[0,1]
[0,1]
It would appear that there is no end to the process of
'crispification': starting with the intuitively well-accepted
canons of classical two-valued logic we may bootstrap ourselves
through the higher reaches of fuzziness. For philosophers, not to
mention computer scientists, this would no doubt be considered a
happy result: a non-classical logic at the foundations of
mathematical reasoning or computation is a forbidding thought
indeed."
>From Chapter IV ("Possibilistic Inductive Inference"):
------------------------------------------------
"Definitions 3 and 4 correspond to Zadeh's (1978) definitions of
possibility distribution function and of possibility distribution
respectively, but differ in an essential way. Zadeh's notion of
the possibility distribution function is regarded as an
interpretation of the notion of membership function of a fuzzy
set, and the two are set equal. The present notion is that the
possibility distribution function represents the relative
possibilities of a set of simple hypotheses generated by the
entire universe of discourse U. Hence Zadeh's definition gives
rise to an absolute function whereas in the present one,
membership functions related by a similarity transformation would
yield the same possibility distribution function -- fall into the
same equivalence class. In short, Zadeh's notion of the
possibility distribution function is absolute, while the present
one is relative. This is all the difference, but it is key: this
is the difference which separates the approximate reasoning
literature from the statistical inference literature. With the
relativist notion of the possibility distribution function, the
gap between the literature on approximate reasoning and that on
statistical inference is bridged, with the likelihood function
serving as the counterpart of the (relativist) possibility
distribution function. This relation will become clearer as we
proceed."
On the "rationality requirements" for inductive logic (from Chap.
IV):
"It will be evident that no _rationality requirements_ have been
laid down for inductive inference, in the manner for example of
Carnap (1971). What a rational man would or should infer about the
value of an unknown when presented with a piece of evidence
bearing on it depends first of all on some truth or reliability
assessment of the speaker uttering the evidence. It seems futile
to attempt to lay down rationality requirements that would
encompass rules for truth assessment in general: what evidence a
person would accept or deny as being 'true' would seem to be more
a matter of behavioral psychology than a matter of pure logic. At
best therefore it would seem that an inductive 'logic' by which a
perfectly rational man may be guided would have to be predicated on
the assumption that truth assessment of evidence is determined
outside of the logic, which then nullifies the whole purpose of
the exercise. This being the case, we take the slightly Popperian
position of taking hypotheses as basic and evaluating these for
truth conditional on truth-assessed evidence (Popper, 1972). What
emerges is the possibility distribution, which provides a way
first of characterizing what a speaker could mean by what he says
taken at face-value, and second of characterizing what a listener
may choose to infer regarding what could be the case after
assessing the evidence for truth/reliability. Thus the possibility
distribution may characterize belief, but it is not a subjective
belief function in the sense of the Bayesians, since at no time is
it necessary to invoke strength or degree of belief as a primitive
attribute for which numeric representation is sought."
On the product-sum rule for evaluation of composite hypotheses
(also from Chap. IV):
"The metaphor of Nature-as-speaker is not too far-fetched when we
consider the frequent use of such expressions as 'What do the data
say...?', or 'What do the data mean ...?' or 'What do the data
tell us...?', and so forth, in reference to the results of
experiment. It seems natural to identify 'the data' with Nature,
in a metaphor very similar to the Bayesians' use of the phrase
'state of Nature' in referring to the true value of the unknown
parameter $ theta $, say, characterizing a probability model. If
we accept the metaphor of 'Nature asserting', then it seems
reasonable to adopt similar meta-semantic considerations for
making inference from assertions of Nature -- the data -- as we do
from speakers in a natural language. Furthermore, if we are to
combine prior assertions of a decision-maker with assertions of
Nature, or the data, it seems some unified interpretive framework
is necessary. This leads us to the conception of the truth or
likelihood of the composite hypothesis { w1, w2 } being identified
with the probability of the calibrational affirmation
'w1 explains the data' \/ 'w2 explains the data'. If we
assume, as we have, that these affirmations should be considered
as independent, or governed by lack of recall from one affirmation
to the next, then we have a product-sum rule for the likelihood
evaluation of composite hypotheses ...
It is conjectured that the product-sum rule for composite
hypotheses should in general resolve adequately the counter-
example used in Chap. I ... to show how the maximum rule for
marginal likelihood may be brought into question."
>From Chapter V ("Possibilistic Deductive Inference"):
-----------------------------------------------
"In this program, the viewpoint is a little different from that of
Zadeh (1977), who saw approximate reasoning as concerned with the
'deduction of possibly imprecise conclusions from a set of imprecise
premises.' Traditional logic, by restricting itself to form rather
than meaning already allows us to reach imprecise conclusions from
imprecise premises, a fact that is exemplified by the example
previously considered:
If one is rich, then one is happy - Premise (Theory 1)
John is rich - Premise (observation)
Therefore, John is happy - Conclusion.
Here both premises are imprecise, as is the conclusion, even
though the rule of deduction which has been applied is quite
exact, relying only on the logical form [(P -> Q) /\ P] -> Q in
the standard logic notation. What we now propose to do is to take
meaning as primary, and to allow deductive inferences to be
validly drawn whenever meaning is preserved, in a sense to be made
clear very shortly. From this viewpoint, the appellation
'approximate' in 'approximate reasoning' would refer not
so much to the rules of logic or of reasoning involved, but to the
nature of the assertions involved, which may in general be fuzzy.
Traditional logic in dealing primarily with form, proceeds almost
entirely on the semantics of form, hardly at all on the semantics
of content. Here we start with content and rules based on form
emerge as a special case."
>From Chapter VI ("Possibilistic Decision Analysis"):
----------------------------------------------
".... But the point is that judgments affecting preference or
choice fall into a wider class of subjective judgments generally,
so that if we could construct methods which apply generally to the
larger problem of subjective estimation or scaling, we would _ipso
facto_ have constructed methods for scaling decision options
(stimuli) on the attribute of level of preference or desirability.
Our approach to the choice problem is to treat it as a scaling
exercise of this sort."
On the scale properties of utility:
"Assumption 1 supposes [the attribute of level of desirability of
a decision option] to possess ratio-scale properties. For those
who are persuaded that utility and therefore the present notion of
level of desirability is at best an interval-scale attribute, a
few remarks appear necessary. A ratio-scale attribute is one whose
universe of discourse could be considered to possess a 'natural'
or 'absolute' zero. This does not appear to be an unreasonable
property for the attribute of level of desirability: the natural
zero corresponds to a desirability level of absolute indifference
-- the situation, say, where one may 'take it or leave it' with
equal equanimity. To one side of the absolute indifference level,
we have positive desirability; to the other side, we have negative
desirability or aversion. This seems to conform to the notion
of weighing the 'pleasures' and 'pains', as Plato put it. This
is such a natural conceptualization of the notion of desirability
(two half-lines, one positive, the other negative with
indifference (= natural zero) in between) that the
fact of the interval-scale characterization of utility in
traditional utility theory is perhaps more the one which needs
examining. What such an examination would reveal is that the use
of interval scale properties for utility represent merely the
weakest assumptions necessary for the purposes of that theory. As
everyone needs to have clarified on first acquaintance with that
theory, 'zero' utility does not necessarily mean 'no'
utility, emphasizing the gap between the natural language
characterization of notions of utility and desirability, and their
approximation within the artificial language of utility theory.
Within that theory, a utility function is defined to be a function
(from a real-world set of rewards to the set of real numbers) such
that given two probability distributions, P1 and P2 (lotteries) on
the set of rewards, P1 is preferred to P2 if and only if the
expectation of the function with respect to P1 is greater than
that with respect to P2 (de Groot 1970, p. 90). This is all we
need if we wish merely to discriminate amongst options, and this
permits arbitrary interval-scale transformations. However it
should not exclude a subjective estimation procedure which
exploits ratio-scale properties. As it turns out, interval-scale
transformations of the final scaling of decision options as to
desirability do not affect choice of the optimum decision option.
This vindicates the utility theory as an acceptable approach in
principle, but it does not invalidate an approach which exploits
ratio-scale properties in the scaling procedure, as we attempt to
do."
On group preferences:
".... Arrow proved that there is no general procedure for
obtaining a group ordering over a set of decision options based on
individual members' preference orderings, that is consistent with
five seemingly reasonable conditions. When utility functions (von
Neumann-Morgenstern type) are used in place of preference
orderings, Harsanyi (1955), and more recently Keeney (1976) have
shown that the impossibility result of Arrow no longer holds. A
group utility function may be constructed using axioms analogous
to Arrow's but stated in terms of utilities rather than preference
orderings. Moreover, the form of the group utility function is
shown to be restricted to the narrow class consisting of linear
combinations of individual utilities, that is of the form W = sum
ki * wi where W is the group utility, wi (i = 1 ,..., n) are
individuals' utilities and ki (i = 1 ,..., n) are constants
reflecting individuals' relative weightings in the aggregation.
The question which this result invites is how do we determine the
ki (i = 1 ,..., n). The discussion by Keeney makes it clear that
a choice of combination weights is essentially a problem of inter-
personal comparisons of utility, moreover it is one which must
devolve around the members of the group as the individuals jointly
responsible for decisions taken under an assumed group utility
function W . Where Assumption 2 represents a different approach
is in side-stepping the aggregation question, and making the
assumption right at the start that individual members have the
capability of judging decision options on the attribute of level
of desirability for the group as a whole. It is therefore
supposed that individual members of the group are capable of the
necessary inter-personal comparisons of satisfaction, though only
in an implicit fashion. This implicitly supposes that each
individual is capable of distinguishing narrow self-interest --
both of himself and of other members -- from the interest of the
group as a whole, and is capable of judging the relative merits,
from the group standpoint, of different compromise solutions when
individual interests must necessarily be in conflict. Without
asking how or why people come to such skills, I would note only
that the harmonious functioning of any group of individuals,
whether as small as a family or large as a nation, seems to depend
upon their existence."
>From Chap. VII ("Bayesian vs. Possibilistic Approaches"):
--------------------------------------------------
"The question raised by all this is just how inevitable is the
probabilistic characterization of prior belief. The Bayesian
explication hinges inordinately on the betting paradigm by which
one's degree of belief in an uncertain proposition is identified
with the odds at which one is willing to bet on it. Thus the
Bayesian argument is only inevitable if the notion of belief is
entirely captured by the operational notion of willingness to bet.
The Bayesians admit, as part of their development, that
willingness to bet on an event reflects not only one's 'degree of
belief' in the possibility of the occurrence of that event, but
also the values one attaches to the stake and potential winnings.
Odds remaining the same, one's willingness to bet one dollar does
not imply willingness to bet 100,000 dollars. Also, belief
remaining the same, one's willingness to bet any given sum changes
quite definitely with the odds offered. The Bayesians fix the
odds by requiring one to be willing either to place or take the
bet, and they take account of value by measuring stakes and
winnings in terms of psychological utility. The remaining
determinant of betting behavior, 'degree of belief', is then
fixed, and simple rules of rationality imply that this belief
function must obey the probability axioms. This is a reductionist
procedure: the notion of belief is that which is left after,
starting with observable betting behavior, we take account of
value and odds, the two other determinants of betting behavior.
Could we not go the other way? What if we were to adopt a
constructionist approach in which the notion of belief is
explicated, built up as it were, from other more primitive
considerations. And what if a constructionist belief
characterization, when joined with consideration of odds and
value, allowed us to explicate betting behavior? The answer, I
think, is that the Bayesian analysis would seem less than
compelling. Furthermore, if such a development preserves the
essential truth of the Bayesian development, that subjective
belief considerations should be incorporated where appropriate,
together with its mathematical convenience, while not requiring
the total commitment to subjectiveness that is an unwelcome
intrusion when we wish to infer strictly on the basis of
experimental evidence, then we have the basis of a compromise that
meets the concerns of both classical and Bayesian schools. Part
of the burden of this essay has been to develop just such an
explication of belief, starting with the semantic underpinnings
provided by a (modified) fuzzy set theory."
>From Chap. VIII ("Summary and Conclusion"):
-------------------------------------
"The end-result of all this is an extended likelihood or
possibility calculus that covers all the Bayesian ground, without
the Bayesian postulate of prior, subjective belief probability.
Prior subjective belief may be linguistically expressed without
need for the Bayesian straitjacket of 'coherence'. Moreover, the
inferential process does not require the injection of prior
evidence or belief, but may accommodate it whenever this is
available, and its inclusion is warranted. Thus the classicists'
major objection to the Bayesian procedure that it needs to suffer
the intrusion, always, of subjective prior opinion may be met,
while the major Bayesian advantage of a direct characterization of
uncertainty of modelling, and an associated powerful calculus is
not sacrificed.
The advantages of a direct characterization of uncertainty of
modelling are reflected most in decision analysis. Bayesian
methods of inference combine quite attractively with an expected
utility approach to decision analysis. The essential structure of
such an approach to decision analysis may be retained here.
However the intrusion of 'fuzz' into probability models does not
conform to the axiomatic development of utility theory, in which
the basic choices are among gambles in which the probabilities are
non-fuzzy. Hence one could either re-develop the traditional
axiomatic utility theory to accommodate fuzz, and its associated
calculus, or one could proceed without utilities. In the latter
case, the outputs of a decision analysis would display the fuzz on
appropriate measures of consequential real-world gain, loss or
risk. Choices could then be made primitively from among competing
possibility distributions on measures of gain or loss. Summary
measures (e.g. center of gravity, area, second moment about the
mean, etc) could of course be employed as necessary.
A benefit of the present approach, by comparison with the Bayesian
approach to decision analysis, is the comparative ease with which
possibilistic prior information may be elicited from decision-
makers. Furthermore, evidence from several sources are treated
symmetrically within the theory, assertions of Nature or
experimental evidence being just one source. This means that a
calculus for the representation of group belief, or the
combination of evidence, and for the aggregation of group
preferences, is easily integrated into the framework.
The development sheds some light, I think, on the Saaty method for
the scaling of judgmental atributes. This essay was motivated in
large part by a desire to explicate the success of the Saaty
method in fuzzy set terms. This led us into the theory of
measurement, and to a realization that the paradigm of point-
numeric measurement needed to be changed to accomodate the idea
engendered by the fuzzy set theory that data are fuzzy sets in
general. This has the salutary consequence that the treatment of
error of measurement is integral to the theory, rather than being
the conceptually difficult and bothersome afterthought that it is
in the now-classical theory exemplified by Krantz, Luce, Suppes,
and Tversky.
The present development also bears on foundational questions of
fuzzy set theory. One of the issues which has been of concern in
the theory has been the relation between, and the line of
demarcation that should separate, the respective applications of
fuzzy and probabilistic calculi. The present development puts
forward an interpretive framework that mixes the two calculi quite
intimately. The somewhat paradoxical result is that the
differences between the two are made very clear, as also are the
respective limits of application of the two concepts. Probability
remains the more basic of the two, and stands in the same relation
to the other, as it already does to the more familiar concept of
likelihood."
Regards,
S.F.Thomas
------------------------------