Hierarchical Structure of Uncertainty
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912023471718400 |
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| author | Adachi, Takanori |
| author_facet | Adachi, Takanori |
| contents | We introduce a new concept called uncertainty spaces which is an extended concept of probability spaces. Then, we express n-layer uncertainty which we call hierarchical uncertainty by a hierarchically constructed sequence of uncertainty spaces, called a U-sequence. We use U-sequences for providing examples that illustrate Ellsberg's paradox. We will use the category theory to get a bird's eye view of the hierarchical structure of uncertainty. We discuss maps between uncertainty spaces and maps between U-sequences, seeing that they form categories of uncertainty spaces and the category of U-sequences, respectively. We construct an endofunctor of the category of measurable spaces in order to embed a given U-sequence into it. Then, by the iterative application of the endofunctor, we construct the universal uncertainty space which may be able to serve as a basis for multi-layer uncertainty theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14219 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hierarchical Structure of Uncertainty Adachi, Takanori Mathematical Finance Probability 91B06, 16B50, 91Gxx, 18C15 We introduce a new concept called uncertainty spaces which is an extended concept of probability spaces. Then, we express n-layer uncertainty which we call hierarchical uncertainty by a hierarchically constructed sequence of uncertainty spaces, called a U-sequence. We use U-sequences for providing examples that illustrate Ellsberg's paradox. We will use the category theory to get a bird's eye view of the hierarchical structure of uncertainty. We discuss maps between uncertainty spaces and maps between U-sequences, seeing that they form categories of uncertainty spaces and the category of U-sequences, respectively. We construct an endofunctor of the category of measurable spaces in order to embed a given U-sequence into it. Then, by the iterative application of the endofunctor, we construct the universal uncertainty space which may be able to serve as a basis for multi-layer uncertainty theory. |
| title | Hierarchical Structure of Uncertainty |
| topic | Mathematical Finance Probability 91B06, 16B50, 91Gxx, 18C15 |
| url | https://arxiv.org/abs/2311.14219 |