A New Definition of Conditional Expectation for Finite Uncertainty Spaces
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 t...
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| Main Authors: | , , |
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| 格式: | text |
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Archīum Ateneo
2024
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| 在線閱讀: | https://archium.ateneo.edu/mathematics-faculty-pubs/261 https://doi.org/10.1063/5.0193426 |
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| 總結: | 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. |
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