Detecting correlation between extreme probability events

ABSTRACT

As classical definitions of correlation give rise to counterintuitive statements when extreme probability events are involved, we introduce enhanced notions of positive and negative correlation in the general framework of coherent conditional probability. These notions allow to handle extreme probability events in a principled way by accommodating the different levels of strength of the zero probabilities involved (namely, zero layers). Since the detection of correlations by means of zero layers is computationally challenging, we provide a full characterization relying on only conditional probability values.

KEYWORDS:

• Conditional probability
• extreme probability event
• coherence
• correlation

Acknowledgements

The authors are members of the GNAMPA-INdAM research group.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 cs is used as an abbreviation of coherent setting.

Additional information

Funding

This research was partially supported by Università degli Studi di Perugia, Fondo Ricerca di Base 2015, project “Inviluppi di probabilità condizionate coerenti e loro applicazioni statistiche”.

Notes on contributors

G. Coletti

Giulianella Coletti is Professor of Probability Theory and Mathematical Statistics in the Department of Mathematics and Computer Science at the University of Perugia (Italy). Her main research interests concern models for managing uncertainty and decisions, also finalized to perform intelligent systems. Moreover, she does research in fuzzy set theory, measurement theory and nonmonotonic logic, with the main aim of “building bridges” between the different theories and developing consistent hybrid inferential tools for knowledge acquisition processes. She has published/coauthored over 200 papers, and is the coauthor of the scientific monograph “Probabilistic Logic in a Coherent Setting” published by Kluwer in the series Trends in Logic in 2002.

L. C. van der Gaag

Linda C. van der Gaag studied Mathematics at Delft University of Technology in the Netherlands. She received her PhD from the University of Amsterdam, the Netherlands, in 1990. After holding assistant and associated professorships at Utrecht University in the Netherlands, she was appointed full professor at the same university in 2000. In that year, she received the prestigious PIONIER grant from the Netherlands Science Foundation; in 2007, she was elected EurAI Fellow by the European Association for Artificial Intelligence. After having served for several years as head of the Department of Information and Computing Sciences at Utrecht University, she joined the Dalle Molle Institute for Artificial Intelligence Research in Lugano, Switzerland, in June 2019. The research interests of Linda van der Gaag are mostly within the field of probabilistic graphical models, and range from their foundations to their application. Her foundational research focuses primarily on probabilistic independence and qualitative concepts of probability, while her applications-oriented work addresses the design of methods for enhancing the practicability of using probabilistic graphical models in the medical and veterinary fields. She published over 200 papers in scientific conferences and journals.

D. Petturiti

Davide Petturiti obtained his PhD in Mathematics and Computer Science for Information and Knowledge Treatment from the University of Perugia, Italy, in 2013. He is currently Assistant Professor of Mathematical Methods of Economics, Finance and Actuarial Sciences in the Department of Economics at the University of Perugia. His main research interests cover decision theory, probability theory, non-additive uncertainty measures, probabilistic and non-probabilistic graphical models, generalized Bayesian inference, stochastic games and gambling systems, imprecise probabilistic models for finance, multi-criteria decision analysis. He published as a coauthor over 40 papers in international journals and conference proceedings.

B. Vantaggi

Barbara Vantaggi is Professor in the Department MEMOTEF at La Sapienza University of Rome (Italy). She received her PhD in Mathematical and Statistical Methods for Economics and Social Sciences from the University of Perugia in 2000. Her research interests are mostly within the field of applied mathematics, uncertainty models, decision theory and graphical models. She coauthored over 100 papers in scientific conferences and journals.