INFT 842: ALTERNATIVE SYSTEMS OF PROBABILISTIC REASONING
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 INFT 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
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.
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 [31 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 [7 - 14 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 [21-28 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
[5 October - 2 November; no class on 12 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
[12 - 23 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 [30 November - 7 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 [INFT 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
Schum, D., Evidence And
Inference For The Intelligence Analyst [Two Volumes],
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.
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
• 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.
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 Argument,
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.
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 which 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.]
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.
von Winterfeldt, D.,
Edwards, W., Decision Analysis And
Lindley, D., Making Decisions, [2nd ed],
• Shafer. G., Savage Revisited, Statistical Science, Vol. 1, No. 4., 1986.
LAST BUT NOT LEAST, WHERE TO FIND YOUR INSTRUCTOR
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: