�������� IT 842: ALTERNATIVE SYSTEMS OF
PROBABILISTIC REASONING
����������������������������������������������� ����� FALL 2005
����������������������������������� ���� INSTRUCTOR: DAVE SCHUM
THE BASIC
SUBJECT MATTER OF THIS COURSE:
����������� In this seminar we will examine
alternative views of the process of drawing conclusions from masses of
incomplete, inconclusive, ambiguous, dissonant, and unreliable evidence. Among
the views we will discuss are the Bayesian,
Baconian, Shafer-Dempster, and Fuzzy
systems for probabilistic reasoning. Each one of these formal systems, as well
as others to be mentioned, has interesting properties that deserve our
attention. The necessity for considering alternative views seems evident since
it seems far too much to expect that any one formal system of probability can
capture all of the richness of probabilistic reasoning. We will examine grounds
for the necessity of considering alternative views of probabilistic reasoning
and, in the process, consider applications of each of these views to
non-trivial inference problems. Of particular interest to us are issues
concerning what seems to constitute "rational" probabilistic
reasoning.
����������� A survey of alternative formal
systems of probabilistic reasoning, by itself, is not quite enough to provide
an adequate background for later applications. Many other issues arise in the
application of any of these formal systems. Some of these issues concern the
fact that most human inference problems seem to involve mixtures of the tasks of inductive, deductive, and what has been termed
"abductive" reasoning. One very important issue concerns the discovery or generation of the
essential ingredients of probabilistic inference: hypotheses/possibilities,
evidence, and arguments providing the linkage between evidence and hypotheses.
Such generation is an exercise in imaginative, creative, or abductive reasoning
[the essential topic of IT 944]. Another important issue concerns the coupling
of probabilistic reasoning and choice. In some contexts [science, for example]
probabilistic reasoning has a life of its own. In many other contexts, however,
such reasoning is embedded in the further process of choice in which difficult
value-related issues arise. We will also examine various contemporary views
about the coupling of probabilities and and values in the process of choice.
TEXTUAL
AND OTHER REFERENCES ON PROBABILISTIC REASONING:
����������� 1). There are many relevant and
interesting works on the topic of probabilistic reasoning to be found in a
variety of disciplines; our lives are simply too short to master all of them.
One trouble is that most published works on the topic of probabilistic
reasoning do not provide much discussion of the very basis for it, namely evidence. Many of the works we
will consider focus just upon algorithms for combining probabilistic judgments
of various sorts and say very little about the evidential foundations for such
judgments. After offering this seminar for a number of years, and at the
suggestion of many students who completed it, I decided to write a book that
covers what I regard as the basic evidential foundations of each of the formal
systems of probability we will discuss. This book is entitled: The Evidential Foundations of Probabilistic
Reasoning� [Northwestern University
Press, Evanston, IL. 2001; paperback edition]. Some of the chapters of this
book are based upon notes I gave students in previous offerings of this course.
In addition, four former students provided editorial and other comments on this
work to ensure that its contents would be especially relevant to the interests
of PhD students in IT&E.
����������� 2). There are, of course, many other
references to consider; these references come from many different disciplines
and frequently involve matters about which there is not widespread awareness.
At the end of this syllabus is a "seed" bibliography containing some
basic references that are keyed to the major probability systems we will
discuss in this seminar.
����������� 3). During this seminar I will
provide you with an assortment of notes and handouts on particular topics. In
most cases these notes will consist of applications of the various
probabilistic reasoning systems and additional sensitivity analyses of the sort
you will find in the text noted in 1) above.
TOPICS AND
READING ASSIGNMENTS
����������� Following is the sequence of topics
we will cover in this seminar together with associated reading assignments
from: Evidential Foundations of
Probabilistic Reasoning [EFPR]. The dates shown are only approximate; we
may wish to dwell longer on some topics than on others.
I. SOME
PRELIMINARY MATTERS [30 August]
����������� To set the stage for later
discussions, I will begin by giving you a bit of history of the development of
interest in the evidential foundations of probabilistic reasoning. In the
process I will try to convince you that any theory of probabilistic reasoning
must be concerned about two major issues: (i) structural matters in the
generation of arguments from evidence to hypotheses/possibilities, and (ii)
matters concerning the assessment and combination of ingredients associated
with the force, strength, or weight of evidence on these
hypotheses/possibilities. Further, I will elaborate a bit on my claim that
human reasoning in natural settings [the "real world", if you like]
involves mixtures of the three forms of reasoning I mentioned above. I will
also give you an overview of the literature that exists in so many disciplines
that is relevant to an understanding of the complexity of probabilistic
reasoning.
II.
EVIDENCE, PROBABILITY, AND STRUCTURAL ISSUES [6 - 13 September].
����������� When most people think about
probability they think only about numbers. But, as I will explain,
probability theories are equally concerned about the arguments that are
constructed to support the use of numbers. The construction of arguments is a
creative task concerning which most of us have very little appropriate
tutoring. This is especially true when we must construct arguments from masses
of evidence. We will consider some exercises showing just how difficult
this can be. One major benefit of studying structural matters is that it allows
us to discern various recurrent forms and combinations of evidence and to
observe how evidence is used during the process of probabilistic reasoning,
regardless of the view of probability one adopts. To be useful in probabilistic
reasoning, evidence must have certain credentials concerning its relevance,
credibility, and inferential force, strength, or weight. No
evidence comes to us with these credentials already established; they must be
established by cogent arguments. The major issue that divides the alternative
views of probabilistic reasoning we will examine involves how best to grade the
inferential force, strength, or weight of evidence and then to combine such
gradings across many items of evidence.
����������� Reading Assignment: EFPR
Chapters 1 - 3.
III.
STRUCTURAL ISSUES AND COMPLEX INFERENCE [20-27 September]
����������� There is presently a substantial
level of research on what are called inference networks. In a wide
variety of applied areas, people have the difficult task of trying to make
sense out of masses of evidence that, upon examination, reveal all of the forms
and combinations of evidence we will have considered in Section II. When
arguments are constructed from a mass of evidence they begin to resemble complex
networks. Many [not all] of the important elements of complex inference can be
captured in graph theoretical terms. Naturally, there are many current attempts
to develop various forms of computer assistance in coping with the complexity
of inference based on masses of evidence. Matters we discuss here will be quite
relevant to other courses now offered in IT&E concerning inference networks
[e.g. Professor Kathy Laskey's INFT 819].
����������� Reading Assignment: EFPR
Chapter 4
IV.
ALTERNATIVE PROBABILITY SYSTEMS I: BASIC THEORETICAL ELEMENTS �
���� [4 October - 8 November; no class on 11
October]
����������� Here we come to a discussion of the
different views about probabilistic reasoning. If only for historical reasons,
we begin with the conventional, Pascalian, or Bayesian system for probabilistic
reasoning. Then we will consider the Shafer-Dempster theory of belief
functions, Cohen's system of Baconian probability for eliminative� and variative induction, and Zadeh's ideas on
fuzzy reasoning and fuzzy probabilities. Our basic objective here is to examine
what one is committed to when one applies any of these formal systems. A
question frequently asked is: which one of these systems do you prefer? This is
rather like asking whether you prefer your saw or your hammer. Each system
"resonates" to particular important elements of probabilistic
reasoning. Each system has something important to tell us about probabilistic
reasoning, but no system says all there is to be said. At this point we will
dwell upon what is meant by the term "rational" probabilistic
reasoning. In this section we will also give attention to the matter of
combining probabilistic and value-related judgments in situations in which
probabilistic reasoning is embedded in the further process of choice.
����������� Reading Assignment: EFPR Chapters
5 - 6
V. ALTERNATIVE
PROBABILITY SYSTEMS II: EXAMPLES AND APPLICATIONS
����������� ���
[15 - 22 November].�
����������� Having examined the theoretical
underpinnings of alternative systems of probabilistic reasoning, we will
examine what each produces in applications in different inferential contexts in
which different forms and combinations of evidence are encountered. In some
cases, our examination will consist of sensitivity analyses performed on
probabilistic expressions designed to capture various forms and combinations of
evidence. It is in this process that we begin to observe how many important and
interesting evidential and inferential subtleties lie just below the surface of
even the "simplest" of probabilistic reasoning tasks. These subtleties,
if recognized, can be exploited in our probabilistic reasoning.
����������� Reading Assignment: EFPR Chapters
7 - 8
VI. DISCOVERING THE INGREDIENTS OF PROBABILISTIC REASONING
[29 November - 6 ��������� Dec.]
����������� As I noted, the ingredients of
probabilistic reasoning are rarely provided for us [except in classroom
exercises and examples]; they have to be discovered or generated. Stated in
other words, there is the necessity for imaginative or creative reasoning in
every probabilistic reasoning task. What we have time to do is simply to
examine some of the imaginative elements of probabilistic reasoning. The topics
of discovery and imaginative reasoning are of sufficient importance that we
devote an entire seminar to the topic [IT 944] that is offered in the Spring
semester.
����������� Reading Assignment: EFPR Chapters
9-10
PROCEDURAL
MATTERS AND METHOD OF EVALUATION
����������� To tell you how I believe this
seminar should proceed, I will make reference to the thoughts of Sir Francis
Bacon. Bacon argued that, in any scholarly activity, reading makes us full, discourse makes us ready, and writing makes us accurate. We will all do a fair amount
of each in this seminar. As far as reading
is concerned, you have one major text and the various handouts and other
materials I will give you. In addition, the reference list I include at the end
of this syllabus contains a wealth of information on all of the probability
systems we will discuss. You can pick and choose among these references
depending upon where your interests take you.
����������� As far as discourse is concerned, I look upon this seminar as an experience
in the sharing of ideas. Indeed, my view of scholarship is that it is a
sophisticated form of sharing; we learn from each other. I have no wish to
monopolize our discussions. On each occasion we meet I will try to get things
started and will bring various matters to your attention. What happens after
this depends upon you. Often, you may believe I am saying outrageous things
with which you disagree; if so, your task is then to show me how I have been
misled. In addition, you may have questions that I have not raised; your task
is to raise them as we proceed. In short, I hope our meetings will be both
enjoyable and productive; whether or not this happens depends upon all of us.��
����������� Finally, the topic of writing brings us to the first of two
methods of evaluating your progress in this seminar. We will encounter
literature from a number of different disciplines; no one discipline dominates
scholarship on probabilistic reasoning. Together, the various works we will
discuss supply a breadth of view. Depth of view is provided by your focus upon
one or more of the formal systems; such focus will form the basis for a paper
you will write, which will count 75% of your grade. The choice of focus
is yours, provided that it concerns probabilistic
reasoning or some matter that directly involves such reasoning. Here are some
characteristics of your written work that I will look for. Notice that I here
mention several criteria that will also be in force as far as your dissertation
is concerned.
����������� (1) Your work should demonstrate
that you have attempted to apply your own thoughts to the issues/problems that
you address. A simple review of literature on some topic [e.g. X says this, Y
says that] will not suffice. What you
think about the issues your paper addresses is what is of primary importance. I
do not insert this criterion out of perversity; it is the most important
criterion that will be applied in evaluating your doctoral research.
����������� (2) Your written work should, of
course, demonstrate awareness of the relevant work of others. When you discuss
previous work on your chosen topic, you should give evidence of having
integrated it in a careful and sensible way and are suitably critical of ideas
you encounter. Unless you do these things, I will not be able to judge how you
drew the conclusions you did from the work of others. This also happens to be a
criterion that will be enforced in your doctoral dissertation.
����������� (3) Perhaps many of you are now
engaged in outside work to which the essential ideas in this seminar are
relevant. In the past I have been quite free in allowing students to report on
their efforts to apply ideas from this seminar to some ongoing project of
interest in their outside work. Sadly, only a small number of these papers have
been of a level of quality one expects from doctoral students. One reason is
that students frequently spend too much time dwelling upon substantive issues
in the problem domain and too little time on the theoretical probabilistic
matters that are being applied in this domain. In some cases there was very
little evidence in a paper that the student actually mastered any probabilistic
matters that could have potential application. I certainly hope and expect that
there will be ideas in this seminar that have applicability in areas of
interest to you. If you do attempt to apply ideas from this seminar to
particular ongoing work you are doing, I will be most concerned about how well
and how completely you have mastered the ideas you are attempting to
apply.��
����������� (4) I will give very careful attention to the coherency with which you present your
ideas. Yes, I am concerned about your ability to communicate your ideas in
written form. I do you no service at all by ignoring grammatical and stylistic
matters.
����������� (5) You should look upon this
written work as something that you do for yourself, not for me. It will simply
demonstrate that you have attempted to acquire some uncommon understanding of a probabilistic inferential issue that
captures your interest.
����������� The remaining 25% of your grade will
be based upon some exercises and problems that I will assign as we proceed. I
expect to give you five such exercises during the semester. Many of these
exercises will involve probabilistic analyses of various forms and combinations
of evidence that appear in your text and in our class discussions.
�
����������� ����������������������������������� A
"SEED" BIBLIOGRAPHY
����������� Following is a list of basic
references to works in each of the alternative systems of probabilistic
reasoning we will discuss. In addition, I have included one or more important
references for each system that are frequently cited but, with less frequency,
actually read.
I.
Views Based Upon The Conventional [Pascalian] Calculus Of Probability.
����������� A.
"Standard" Bayesianism
����������� Perhaps
the most frequently cited but infrequently read item on our list is the
original paper by Thomas Bayes that was communicated by Richard Price to the
Royal Society Of London two years after Bayes' death.
1) An
Essay Towards Solving A Problem In The Doctrine Of Chances. By The Late
Rev. Mr Bayes, F. R. S. Communicated by Mr. Price in a letter to John Canton,
A. M., F. R. S., Philosophical
Transactions Of The Royal Society, pp 370-418, 1763.
�����������
����������� There
is now debate about whether or not what has become known as Bayes' Rule should,
in fact, be attributed to Bayes. Here is a paper on the idea that others may
have earlier tumbled to the essence of Bayes' Rule.
2) Stigler, S. M., Who Discovered
Bayes's Theorem ? The American
Statistician, November 1983, Vol. 37, No. 4, p 290 - 296.
����������� Here
are some tutorials on the use of Bayes' rule in probabilistic inference.
3) von Winterfeldt, D., Edwards, W., Decision Analysis And Behavioral Research,
Cambridge, Cambridge University Press, 1986, Chapters 5 and 6.
4) Schum, D., Evidence And Inference For The Intelligence Analyst [Two Volumes],
Lanham, Md., University Press Of America, 1987. Chapters 6, 7, and their
supplements.
����������� There
are now many references relative to the use of Bayes' Rule in statistical inference. Here are
two places to start if you have interest in statistical inference.
5) Winkler, R., Introduction To Bayesian Inference And Decision, New York,
Holt, Rinehart, & Winston, 1972.
6) O'Hagen, A., Probability: Methods And Measurement, London, Chapman & Hall,
1988.
����������� Here
are three works that, to varying degrees, consider Bayes' rule as the
normative standard for probabilistic reasoning:
9) Good, I. J. Good Thinking: The Foundations of Probability and Its Applications.
University of Minnesota Press, Minneapolis. MN, 1983
8) Earman, J., Bayes or Bust ?: A Critical Examination of Bayesian Confirmation Theory,
Bradford Books, MIT Press, 1992
9) Howson, C., Urbach, P., Scientific Reasoning: The Bayesian Approach,
Open Court Press, 1989.
����������� Here
are some works on the application of Bayes' rule in complex "inferential
networks"
10) Pearl, J., Probabilistic Reasoning In Intelligent Systems: Networks Of Plausible
Inference, San Mateo Calif., Morgan, Kaufman Publishers, 1988
11) Schum, D., Evidence And Inference For The Intelligence Analyst [Two Volumes],
Lanham, Md., University Press Of America, 1987. Particularly Chapters 4, and 8
through 12 and their supplements.
12) Kadane, J., Schum, D. A Probabilistic Analysis of the Sacco and
Vanzetti Evidence. New York, Wiley, 1996 [A Bayesian analyses of the
evidence in a celebrated murder trial]
13) Glymour, C. The Mind's Arrows: Bayes Nets and Graphical Causal Models in Psychology.
MIT Press, Cambridge, MA, 2001
����������� We
should note here that there is a very large literature on various possible
interpretations of numbers that correspond to the standard or Pascalian
calculus. Indeed, this is an entire topic by itself. There are two useful
summaries of alternative conceptions of a Pascalian probability, they are:
13)�
Fine, T., Theories Of
Probability: An Examination Of Foundations, New York, Academic Press, 1973
14) Weatherford, R., Philosophical Foundations of Probability
Theory, London, Routledge & Kegan Paul, 1982
����������� B.
"Non-Standard" Bayesianism.
����������� It
has been claimed by several Scandinavians [and one renegade Englishman] that we
are frequently misguided in the manner in which we apply Bayes' rule in drawing
conclusions from evidence. They have put together what is described as the evidentiary value model [EVM].
This model is basically Bayesian but it rests upon a particular view of
evidence and how it should be used. I include EVM on this listing because it
provides an interesting transition to the next major view, the non-additive
system of beliefs proposed by Glenn Shafer. Over 20 years ago, I challenged the
EVMers to show how this system is in any way congenial to application in
cascaded or hierarchical inference; I am still waiting for a reply. Here is the
major reference to EVM research:
15) Gardenfors, P., Hansson, B.,
Sahlin, N. E. [eds], Evidentiary Value:
Philosophical, Judicial, And Psychological Aspects Of A Theory. Lund,
Sweden, C. W. K. Gleerups, 1983.
����������� There
are several references in the above work that are relevant to our next major
view of probabilistic inference.
II.
A Theory Of Nonadditive Beliefs Based On Evidence.
����������� It
is easily shown how probabilities encountered in aleatory [games of chance] and frequentistic
[statistical] contexts can be trapped within the Pascalian calculus. But what
about the array of epistemic contexts
in which numbers are used to grade the strength
of our beliefs about whether or not some event has happened, is happening,
or will happen ? In such contexts we often encounter singular, unique, or
nonreplicable events that can have no frequentistic interpretation. In
epistemic contexts many commonly-encountered credal or belief states cannot
easily be trapped within the bounds of the�
Pascalian system; it turns out that this has been known for centuries.
Our first reference here is a very useful and well-done treatise on the history
of the concept of probability. Discussed in this treatise are some of the
difficulties the Pascalian calculus has experienced in the trapping of these
credal states.
1) Hacking, I., The Emergence Of Probability, Cambridge, Cambridge University
Press, 1978.
����������� There
are alternatives to the use of Bayes' rule in the task of combining our beliefs
based on some emerging body of evidence; this has also been recognized for
centuries. One mechanism for belief combination has been termed
"Dempster's rule"; this rule has roots in much earlier work. In 1976
Glenn Shafer took Dempster's rule as the cornerstone for a "new"
system of probabilistic reasoning involving what he terms a "belief function".
We shall term this species of probabilistic reasoning the
"Shafer-Dempster" system.
����������� A.
The Shafer-Dempster View.
����������� �Shafer was a student of Dempster's and has now
achieved a considerable measure of fame as a result of the following work:
2)�
Shafer, G., A Mathematical Theory
Of Evidence, Princeton, N.J., Princeton University Press, 1976.
����������� Some
of us believe this work is not actually a theory of evidence but a theory of
belief based upon evidence; form your own opinion as we discuss his work and
the work of others. Here is a summary of recent thinking about the
Shafer-Dempster system of belief functions:
3) Yager, R., Fedrizzi, M., Kacprzyk,
J. Advances in the Dempster-Shafer
Theory of Evidence. Ney York,
John Wiley & Sons, 1994.
����������� Here
are several often cited works by Glenn Shafer that elaborate on the historical
foundations of the Shafer-Dempster view and that also are critical of any view
in which Bayes' rule is advocated as the
"normative" or "prescriptive" view of probabilistic
reasoning.
4) Shafer, G., Bayes's Two Arguments
For The Rule Of Conditioning, The Annals
Of Statistics, Vol. 10, No. 4, 1982.
5) Shafer, G., Conditional Probability,
International Statistical Review,
Vol. 53, 1985
6)Shafer, G., The Combination Of
Evidence, International Journal Of
Intelligent Systems, Vol. 1, 1986.
7) Shafer, G., The Construction of
Probability Argument, Boston University
Law Review, Vol. 66, Nos, 3 & 4, May/July, 1986.
8) Shafer, G., Pearl, J., (eds), Readings in Uncertain Reasoning, Morgan
Kaufmann Publishers, 1989 [compares Bayes with Shafer-Dempster. Formerly
assigned as a text in this seminar].
����������� B.
"Potential Surprise": Another Nonadditive System.
����������� More
than one person has been interested in relationships between the concepts of probability and possibility; one such person is the British economist G. L. S.
Shackle. In the following work Shackle proposed a metric, called potential surprise, for grading our
beliefs about the possibility of events, something he believed was not possible
within the� Pascalian calculus.
9) Shackle, G. L. S., Decision, Order, And Time In Human Affairs,
Cambridge, 1969.�
����������� This
system is still being discussed, thanks to the efforts of the American
philosopher Isaac Levi. Levi claims this system to be quite general and sees in
it some parallels with the Shafer-Dempster system. In some quarters this system
of reasoning is referred to as the Levi-Shackle system of reasoning. Here are
some further, more up to date references to potential surprise.
10) Levi., I., Decision And Revision: Philosophical Essays On Knowledge And Value,
Cambridge, Cambridge University Press, 1984 [Chapter 14].
11) Levi, I.,� The
Enterprise Of Knowledge: An Essay On Knowledge, Credal Probability, And Chance.
Boston, MIT Press, 1983
III.
Probability In Eliminative and Variative Induction.
����������� In
many contexts, science for example, we subject our hypotheses to a testing
process in which only the fittest survive. In such tests evidence is used as a
basis for eliminating hypotheses. As
Professor L. Jonathan Cohen will tell us in his book, a particular hypothesis
seems to have increasing probability or provability as it survives our best
efforts to invalidate it. We subject hypotheses to a variety of different
evidential tests; the more of these tests some hypothesis survives, the more confidence
we have in it. The key word here is variety;
the survival of any hypothesis depends upon the extent to which it holds up
under different conditions.
Replication of test results is important but we cannot gather support for some
hypothesis simply by performing the same test over and over again. Drawing upon
the work of Sir Francis Bacon and John Stuart Mill, Professor Cohen has given
us a system of probability that is suited to what he terms eliminative and variative
inductive inference. In this system of "Baconian" probabilistic
reasoning, probabilities grade the extent to which some hypothesis survives an
eliminative testing process, The "weight" of evidence, in Baconian
terms, is related to the number of
evidential tests we perform and to the extent to which our tests cover
variables relevant in discriminating among the hypotheses we consider. Cohen's
system is the only system that specifically grades the completeness or the sufficiency of our evidence. How likely we view
some hypothesis depends upon how many relevant questions concerning our
hypotheses that our existing evidence does not answer.
����������� The
first reference is to Cohen's work on developing means for grading the inductive support that evidence
provides in the eliminative testing process Cohen describes.
1) Cohen, L. J., The Implications Of Induction, London, Methuen & Co. Ltd.,
1970.
����������� Cohen
certainly acknowledges that there is room for more than one view of
probabilistic reasoning. Cohen's "polycriterial" account of
probability is given in the following three references, the second of which can
probably be termed his major work.
2) Cohen. L. J., Probability: The One
And The Many, Proceedings Of The British
Academy, V ol. LXI, 1975.
3) Cohen. L. J., The Probable And The Provable, Oxford, The Clarendon Press, 1977.
4) Cohen, L. J., An Introduction to the Philosophy of Induction and Probability,
Oxford University Press, 1989. I used to require every student to read this
work; it is truly outstanding. Unfortunately, it is now out of print.
����������� Here
are several other of Cohen's works that are of particular importance to anyone
seeking to understand the full dimensions of Cohen's views.
5) Cohen, L. J., Bayesianism Versus
Baconianism In The Evaluation Of Medical Diagnosis. British Journal For The Philosophy Of Science, Vol. 31, 1980.
6) Cohen, L. J., Twelve Questions About
Keynes's Conception Of Weight, British
Journal For The Philosophy Of Science, Vol. 37, 1985.
7) Cohen, L. J., Hesse, M., The Applications Of Inductive Logic,
Oxford, The Clarendon Press, 1980.
8) Cohen, L. J., The Dialogue Of Reason, Oxford, The Clarendon Press, 1986.
����������� Here
are three references in which Cohen's Baconian system is discussed and compared
with other views.
9) Schum, D., A Review Of A Case
Against Blaise Pascal And His Heirs, University
Of Michigan Law Review, Vol. 77, No. 3, 1979.
10) Schum, D., Probability And The
Processes Of Discovery, Proof, And Choice. Boston
University Law Review, Vol. 66., Nos. 3 & 4, May/July 1986.
11) Schum, D., Jonathan Cohen And
Thomas Bayes On The Analysis Of Chains Of Reasoning. In: Rationality And Reasoning: Essays In Honor Of L. Jonathan Cohen
[eds. Eells, E., Maruszewski, T.] Amsterdam, Rodopi, 1991.
IV.
Imprecision, Fuzzy Probabilities, And Possibilities.
����������� The
Pascalian system of probability is rooted in a system of two-valued logic; a
statement is either true or false or a particular element is either in some
subset or it isn't. In 1965 Professor Lotfi Zadeh argued that our inferences
and decisions are often based upon information that is imprecise or abiguous and for which this two-valued logic is
inappropriate. He coined the term "fuzzy sets" to describe
collections of elements with indistinct, imprecise, or "elastic"
boundaries. Zadeh has argued that a different calculus is necessary to
represent reasoning based upon fuzzy information. His early work has generated
enormous enthusiasm and there are now over 15,000 papers, books, and other
materials that have been published on fuzzy matters since 1965, the year in
which Zadeh's work on fuzzy sets saw the light of day.
1) Zadeh, L., "Fuzzy Sets", Information And Control, Vol. 8, 1965.
����������� Zadeh
and his now enormous international body of followers have published papers on
the application of fuzzy sets in an array of inferential and decisional
contexts. The best single collection of Zadeh's papers is found in the
following.
2) Yager, R., Ovchinnikov, S., Tong,
R., Nguyen, H., Fuzzy Sets And
Applications: Selected Papers By L. A. Zadeh, New York, Wiley & Sons,
1987.
����������� Here
are several fairly current works that may be regarded as tutorial regarding
fuzzy sets and systems.
3) Klir, G., Folger, T., Fuzzy Sets, Uncertainty, and Information,
Prentice Hall, 1988
4) McNeill, D., Freiberger, P., Fuzzy Logic, Simon & Schuster, 1993
5)Kosko, B., Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, 1993.
This is an absorbing (but frequently irritating) work by the leading
spear-carrier of fuzzy reasoning.
����������� The
following work is critical of the idea that fuzzy logic actually extends
classical logic.
6) Haack, S. Deviant Logic, Fuzzy Logic: Beyond the Formalism. University of
Chicago Press, 1996.��
V.
Probabilistic Inference And Its Role In Decisions
����������� As
noted in your syllabus, we should give attention to inferential activity in the
following two contexts. In many situations, various areas of science for
example, inferential activity is simply part of the process of
knowledge-acquisition. However, in other situations such as in law, medicine,
intelligence analysis, and so on, inferential activity is embedded in the
further process of choice. In all of these other situations, assessments of
probability have somehow to be combined with assessments of the value of
consequences that occur to us when we contemplate various choices in the face
of uncertainty. There is a substantial literature on the combination of
inferential and value-related ingredients in choice under uncertainty. Two
quite different views of this process of combination are found in the following
three references.
1) von Winterfeldt, D., Edwards, W., Decision Analysis And Behavioral Research,
Cambridge, Cambridge University Press, 1986.�
[See Chapters 1, 2, 3]
2) Lindley, D., Making Decisions, [2nd ed], London, Wiley & Sons, 1985.
3) Shafer. G., Savage Revisited, Statistical Science, Vol. 1, No. 4.,
1986.
LAST
BUT NOT LEAST, WHERE TO FIND YOUR INSTRUCTOR
����������� I
usually lurk in the vicinity of Room 111-A, Science & Technology Bldg. II;
my office phone is 703-993-1694. If I am not in my office, I am almost
certainly at home: 2219 Chestertown Dr., Vienna, Va. My home phone is:
703-698-9515. Please do not hesitate to contact me at any of these locations;
come by my office any time the mood suits, you are always welcome. If all else
fails, or if you simply prefer electronic mail, my e-mail address is:
<[email protected]>. My home e-mail is:
<[email protected]>.� I will
do all I can to make this seminar useful and enjoyable for you.
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