A New Definition of Conditional Expectation for Finite Uncertainty Spaces

• Main Authors: Frondoza, Michael, Eden, Richard, De Lara-Tuprio, Elvira
• Format: text
• Published: Archīum Ateneo 2024
• Subjects: atoms of a σ-algebra; Choquet integral; conditional expectation; uncertain measure; Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/261; https://doi.org/10.1063/5.0193426
• Institution: Ateneo De Manila University

This paper continues the authors' previous work on developing a theory of conditional expectations in uncertainty spaces. In a previous paper, they adopted the standard definition from classical probability by defining the conditional expectation E[X|G] of an uncertain variable X with respect to a σ-algebra G as a G-measurable function provided by a version of the Radon-Nikodym Theorem for uncertainty spaces. In this current work, a definition is provided by minimizing the expected mean squared error (X .Y)2 among G -measurable functions Y. The development, adopted from an existing work on non-additive probability spaces and repurposed for the current setting, similarly assumes a finite sample space and hence finitely many atoms for G. It also justifies the existence of conditional expectations and discusses some of their properties.