A Theory of Uncertainty Variables for State Estimation and Inference
© 2019 IEEE. Probability theory forms an overarching framework for modeling uncertainty, and by extension, also in designing state estimation and inference algorithms. While it provides a good foundation to system modeling, analysis, and an understanding of the real world, its application to algorit...
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| Format: | Article |
| Language: | English |
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IEEE
2021
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| Online Access: | https://hdl.handle.net/1721.1/136725 |
| _version_ | 1826215245478100992 |
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| author | Talak, Rajat Karaman, Sertac Modiano, Eytan |
| author2 | Massachusetts Institute of Technology. Laboratory for Information and Decision Systems |
| author_facet | Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Talak, Rajat Karaman, Sertac Modiano, Eytan |
| author_sort | Talak, Rajat |
| collection | MIT |
| description | © 2019 IEEE. Probability theory forms an overarching framework for modeling uncertainty, and by extension, also in designing state estimation and inference algorithms. While it provides a good foundation to system modeling, analysis, and an understanding of the real world, its application to algorithm design suffers from computational intractability. In this work, we develop a new framework of uncertainty variables to model uncertainty. A simple uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set map, from a given realization of a variable to a set of possible realizations of another variable. We prove Bayes' law and the law of total probability equivalents for uncertainty variables. We define a notion of independence, conditional independence, and pairwise independence for a collection of uncertainty variables, and show that this new notion of independence preserves the properties of independence defined over random variables. We then develop a graphical model, namely Bayesian uncertainty network, a Bayesian network equivalent defined over a collection of uncertainty variables, and show that all the natural conditional independence properties, expected out of a Bayesian network, hold for the Bayesian uncertainty network. We also define the notion of point estimate, and show its relation with the maximum a posteriori estimate. |
| first_indexed | 2024-09-23T16:20:04Z |
| format | Article |
| id | mit-1721.1/136725 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:20:04Z |
| publishDate | 2021 |
| publisher | IEEE |
| record_format | dspace |
| spelling | mit-1721.1/1367252023-06-28T18:25:08Z A Theory of Uncertainty Variables for State Estimation and Inference Talak, Rajat Karaman, Sertac Modiano, Eytan Massachusetts Institute of Technology. Laboratory for Information and Decision Systems © 2019 IEEE. Probability theory forms an overarching framework for modeling uncertainty, and by extension, also in designing state estimation and inference algorithms. While it provides a good foundation to system modeling, analysis, and an understanding of the real world, its application to algorithm design suffers from computational intractability. In this work, we develop a new framework of uncertainty variables to model uncertainty. A simple uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set map, from a given realization of a variable to a set of possible realizations of another variable. We prove Bayes' law and the law of total probability equivalents for uncertainty variables. We define a notion of independence, conditional independence, and pairwise independence for a collection of uncertainty variables, and show that this new notion of independence preserves the properties of independence defined over random variables. We then develop a graphical model, namely Bayesian uncertainty network, a Bayesian network equivalent defined over a collection of uncertainty variables, and show that all the natural conditional independence properties, expected out of a Bayesian network, hold for the Bayesian uncertainty network. We also define the notion of point estimate, and show its relation with the maximum a posteriori estimate. 2021-10-28T18:07:51Z 2021-10-28T18:07:51Z 2019-09 2021-04-30T17:20:47Z Article http://purl.org/eprint/type/ConferencePaper https://hdl.handle.net/1721.1/136725 Talak, Rajat, Karaman, Sertac and Modiano, Eytan. 2019. "A Theory of Uncertainty Variables for State Estimation and Inference." 2019 57th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2019. en 10.1109/allerton.2019.8919919 2019 57th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2019 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf IEEE arXiv |
| spellingShingle | Talak, Rajat Karaman, Sertac Modiano, Eytan A Theory of Uncertainty Variables for State Estimation and Inference |
| title | A Theory of Uncertainty Variables for State Estimation and Inference |
| title_full | A Theory of Uncertainty Variables for State Estimation and Inference |
| title_fullStr | A Theory of Uncertainty Variables for State Estimation and Inference |
| title_full_unstemmed | A Theory of Uncertainty Variables for State Estimation and Inference |
| title_short | A Theory of Uncertainty Variables for State Estimation and Inference |
| title_sort | theory of uncertainty variables for state estimation and inference |
| url | https://hdl.handle.net/1721.1/136725 |
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