LIDS Colloquium

Limitations of Standard Probability and Some Alternatives

Terrence L. Fine

Cornell University

Tuesday, Oct 2nd, 2007
4:00-5:00 p.m.
32-141

A short reception will follow on the 6th floor of the Dreyfoos Building.

Abstract:
We tour the realm of probability so as to survey the disputed and indeterminate boundary between so-called standard numerically-definiteprecise and imprecise probability. The issue is less that of precision versus imprecision than that of the choice of appropriately expressive mathematical models for a wide range of probabilistic phenomena. A mathematical model is only “imprecise” when it is insufficiently expressive and therefore unable to capture essential aspects of the phenomena under study. It is a long-standing prejudice that there is only scientific knowledge when there is recourse to number. In the realm of probability (especially subjective probability) there are core concepts and applications that do not admit of the meaningful ultra-fine distinctions offered by the real numbers. It has even been argued that the real numbers are insufficiently expressive and one might need to resort to the non-standard reals. We will balance our critical remarks by providing several alternative mathematical probability models that have received far less attention but that offer models that better fit certain probabilistic phenomena.

I hope that this tour of probability will clarify the importance of: starting from meaning, often as found either through ordinary usage and common experience and reflection thereon or through scientific investigation of phenomena removed from everyday experience; selection of an appropriately expressive mathematical model, perhaps real-valued set functions but not universally so; and lastly ensuring that our subsequent axiomatization, use of de?nitions to capture important issues, and their implications exposed through mathematical investigations do not lose sight of their origins in probabilistic reasoning. For many of us the attraction of mathematical investigation bids us follow

lines of thought that have little relationship to our original motivations. John von Neumann has been perspicuous in this, as in much else. The work of Patrick Suppes is the most insightful and sustained on the issue of relations between the realms of probability and those of mathematical models.

Biography:

Terrence L. Fine is Associate Director of the School of Electrical and Computer Engineering at Cornell University and former Director of the Center for Applied Mathematics at Cornell. He is also a member of the Cornell Department of Statistical Science and of the graduate fields of Applied Mathematics, Electrical Engineering, and Statistics. He has twice been a visiting professor at Stanford.

Fine was the last president of the IEEE Information Theory Group and has been an associate editor for detection and estimation and for book reviews of the Transactions. He has been a member of the governing board of the IEEE Neural Networks Council, a founding member and director of the NIPS Foundation, and is currently a member of the recently-created Executive Committee of the Society for Imprecise Probability: Theory and Applications (SIPTA).

Fine is the author of the monographs, Theories of Probability: An Examination of Foundations, Academic Press, 1973, and Feedforward Neural Network Methodology, Springer, 1999, and the textbook, Probability and Probabilistic Reasoning for Electrical Engineering, Prentice Hall, 2006.

His enduring interests are in the foundations of probability understood as possible meanings or interpretations of probability and alternative mathematical concepts of probability.