IT 842: ALTERNATIVE SYSTEMS OF PROBABILISTIC REASONING
FALL 2003
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 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, 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. Make sure that you get the paperback version of this
text. It is also available in a hardback version from John Wiley & Sons
[1994 ed]. Wiley charges an absolutely larcenous price
for this hardback edition.
����������� 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
����������� 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 [27
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 [3 -101 Sept].
����������� 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 [17 - 24 Sept.]
����������� 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.
����������� Reading
Assignment: EFPR Chapter 4
IV. ALTERNATIVE PROBABILITY
SYSTEMS I: BASIC THEORETICAL ELEMENTS �
����
[1 - 29 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
����������� ��� [5 - 12 Nov].�
����������� 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 [19 November - 3 ��������� Dec.� No class on 26 November]
����������� 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.
����������� �
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.
����������� �
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.
����������� �
von Winterfeldt, D.,
Edwards, W., Decision Analysis And
Behavioral Research,
����������� �
Schum, D., Evidence And
Inference For The Intelligence Analyst [Two Volumes],
����������� There
are now many references concerning the use of Bayes'
Rule in statistical inference. Here
are two places to start if you have interest in statistical inference.
����������� �
Winkler, R., Introduction To Bayesian Inference And Decision,
����������� �
O'Hagen, A., Probability:
Methods And Measurement,
����������� Here
are two works that, to varying degrees, consider Bayes'
rule as the normative standard for probabilistic reasoning:
����������� �
Earman, J., Bayes or Bust ?: A Critical Examination of Bayesian Confirmation
Theory,
����������� �
Howson, C., Urbach, P., Scientific Reasoning: The Bayesian Approach,
Open Court Press, 1989.
����������� Here
are four works on the application of Bayes' rule in
complex "inferential networks"
����������� �
����������� �
Schum, D., Evidence And
Inference For The Intelligence Analyst [Two Volumes],
����������� �
Neapolitan, R. E., Probabilistic
Reasoning In Expert Systems,
����������� �
Kadane, J., Schum, D. A Probabilistic Analysis of the Sacco and Vanzetti Evidence.
����������� 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:
����������� �
Fine, T., Theories Of
Probability: An Examination Of Foundations,
����������� �
Weatherford, R., Philosophical
Foundations of Probability Theory,
����������� 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 a dozen 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:
����������� �
Gardenfors, P., Hansson,
B., Sahlin, N. E. [eds], Evidentiary
Value: Philosophical, Judicial, And Psychological Aspects Of A Theory.
����������� 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.
����������� �
Hacking,
����������� 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:
����������� �
Shafer, G., A Mathematical Theory Of Evidence,
����������� 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:
����������� �
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.
����������� �
Shafer, G., Bayes's Two Arguments For The Rule Of
Conditioning, The Annals Of Statistics,
Vol. 10, No. 4, 1982.
����������� �
Shafer, G., Conditional Probability, International
Statistical Review, Vol. 53, 1985
����������� �
Shafer, G., The Combination Of Evidence, International Journal Of Intelligent
Systems, Vol. 1, 1986.
����������� �
Shafer, G., The Construction of Probability Arguments,
����������� �
Shafer, G.,
����������� 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.
����������� �
Shackle, G. L. S., Decision, Order, And Time In Human Affairs,
����������� 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.
����������� �
Levi.,
����������� �
Levi,
III. Probability And Eliminative 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.
����������� �
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.
����������� �
Cohen. L. J., Probability: The One And The Many, Proceedings Of The British Academy, V ol. LXI, 1975.
����������� �
Cohen. L. J., The Probable And The Provable,
����������� �
Cohen, L. J., An Introduction to the Philosophy of
Induction and Probability,
����������� 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.
����������� �
Cohen, L. J., Bayesianism Versus
Baconianism In The Evaluation Of Medical Diagnosis. British Journal For
The Philosophy Of Science, Vol. 31, 1980.
����������� �
Cohen, L. J., Twelve Questions About Keynes's
Conception Of Weight, British Journal
For The Philosophy Of Science, Vol. 37, 1985.
����������� �
Cohen, L. J., Hesse, M., The Applications Of Inductive Logic,
����������� �
Cohen, L. J., The Dialogue Of Reason,
����������� Here
are three references in which Cohen's Baconian system
is discussed and compared with other views. I have copies of these works that
you are welcome to have.
����������� �
Schum, D., A Review Of A Case Against Blaise Pascal
And His Heirs, University Of Michigan
Law Review, Vol. 77, No. 3, 1979.
����������� �
Schum, D., Probability And The Processes Of Discovery,
Proof, And Choice.
����������� �
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.]
����������� Here,
finally, is very recent collection of Jonathan Cohen's papers, many of which
concern his thoughts on probability:
����������� Cohen,
L. J. Knowledge and Language: Selected Essays of L. Jonathan Cohen. Ed.
J. Logue. Kluwer Academic Publishers,
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.
����������� �
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.
����������� �
Yager, R., Ovchinnikov, S.,
Tong, R., Nguyen, H., Fuzzy Sets And Applications: Selected Papers By L. A. Zadeh,
����������� Here
are several current works that may be regarded as tutorial regarding fuzzy sets
and systems.
����������� �
Klir, G., Folger, T., Fuzzy Sets, Uncertainty, and Information,
Prentice Hall, 1988
����������� �
McNeill, D., Freiberger, P., Fuzzy Logic, Simon & Schuster, 1993
����������� �
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.
����������� �
Haack, S. Deviant
Logic, Fuzzy Logic: Beyond the Formalism.
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.
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von Winterfeldt, D.,
Edwards, W., Decision Analysis And
Behavioral Research,
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Lindley, D., Making Decisions, [2nd ed],
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Shafer. G., Savage Revisited, Statistical
Science, Vol. 1, No. 4., 1986.
LAST BUT NOT LEAST, WHERE TO FIND YOUR INSTRUCTOR
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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:
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