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A new approach for imprecise probabilities

Marcello Basili, Luca Pratelli

Abstract

This paper introduces a novel concept of interval probability measures that enables the representation of imprecise probabilities, or uncertainty, in a natural and coherent manner. Within an algebra of sets, we introduce a notion of weak complementation denoted as $ψ$. The interval probability measure of an event $H$ is defined with respect to the set of indecisive eventualities $(ψ(H))^c$, which is included in the standard complement $H^c$. We characterize a broad class of interval probability measures and define their properties. Additionally, we establish an updating rule with respect to $H$, incorporating concepts of statistical independence and dependence. The interval distribution of a random variable is formulated, and a corresponding definition of stochastic dominance between two random variables is introduced. As a byproduct, a formal solution to the century-old Keynes-Ramsey controversy is presented.

A new approach for imprecise probabilities

Abstract

This paper introduces a novel concept of interval probability measures that enables the representation of imprecise probabilities, or uncertainty, in a natural and coherent manner. Within an algebra of sets, we introduce a notion of weak complementation denoted as . The interval probability measure of an event is defined with respect to the set of indecisive eventualities , which is included in the standard complement . We characterize a broad class of interval probability measures and define their properties. Additionally, we establish an updating rule with respect to , incorporating concepts of statistical independence and dependence. The interval distribution of a random variable is formulated, and a corresponding definition of stochastic dominance between two random variables is introduced. As a byproduct, a formal solution to the century-old Keynes-Ramsey controversy is presented.
Paper Structure (14 sections, 4 theorems, 77 equations)

This paper contains 14 sections, 4 theorems, 77 equations.

Table of Contents

  1. Introduzione
  2. Weak and ${\mathcal{Z}}$-complementation
  3. Interval Probability Measures
  4. Conditional Interval Probability Measures
  5. Independence of an event $A$ conditioned to $H$
  6. Interval Distribution of a Random Variable
  7. A possible solution of controversy between Key-nes and Ramsey.
  8. Conclusion
  9. Appendix
  10. Proof Proposition 4
  11. Proof of (11)
  12. Proof Proposition 13
  13. Proof Proposition 14
  14. Proof Proposition 18

Key Result

Proposition 4

$\psi_{\mathcal{Z}}$ is a regular weak complementation and, for any pair $H,K$ of events, the following identities are true: Moreover, and where $J=\cup_{n: H\cap Z_n \neq \emptyset=K\cap Z_n}\, H^c\cap Z_n \, \cup_{n: K\cap Z_n \neq \emptyset=H\cap Z_n}\, K^c\cap Z_n.$ In particular

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Remark 3
  • Proposition 4
  • Example 5
  • Definition 6
  • Definition 7
  • Example 8
  • Definition 9
  • Definition 10
  • ...and 11 more