Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure
This work proposes an optimization formulation to determine a set of empirical importance weights to achieve a change of probability measure. The objective is to estimate statistics from a target distribution using random samples generated from a (different) proposal distribution. This work consider...
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| Format: | Article |
| Language: | English |
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Springer US
2017
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| Online Access: | http://hdl.handle.net/1721.1/107480 https://orcid.org/0000-0003-2156-9338 https://orcid.org/0000-0001-8410-6141 |
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| author | Allaire, Douglas Willcox, Karen E Amaral, Sergio Daniel Marques |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Allaire, Douglas Willcox, Karen E Amaral, Sergio Daniel Marques |
| author_sort | Allaire, Douglas |
| collection | MIT |
| description | This work proposes an optimization formulation to determine a set of empirical importance weights to achieve a change of probability measure. The objective is to estimate statistics from a target distribution using random samples generated from a (different) proposal distribution. This work considers the specific case in which the proposal distribution from which the random samples are generated is unknown; that is, we have available the samples but no explicit description of their underlying distribution. In this setting, the Radon–Nikodym theorem provides a valid but indeterminable solution to the task, since the distribution from which the random samples are generated is inaccessible. The proposed approach employs the well-defined and determinable empirical distribution function associated with the available samples. The core idea is to compute importance weights associated with the random samples, such that the distance between the weighted proposal empirical distribution function and the desired target distribution function is minimized. The distance metric selected for this work is the L[subscript 2] -norm and the importance weights are constrained to define a probability measure. The resulting optimization problem is shown to be a single linear equality and box-constrained quadratic program. This problem can be solved efficiently using optimization algorithms that scale well to high dimensions. Under some conditions restricting the class of distribution functions, the solution of the optimization problem is shown to result in a weighted proposal empirical distribution function that converges to the target distribution function in the L[subscript 1] -norm, as the number of samples tends to infinity. Results on a variety of test cases show that the proposed approach performs well in comparison with other well-known approaches. |
| first_indexed | 2024-09-23T15:53:07Z |
| format | Article |
| id | mit-1721.1/107480 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:53:07Z |
| publishDate | 2017 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1074802022-09-29T16:48:33Z Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure Optimal L2-norm empirical importance weights for the change of probability measure Allaire, Douglas Willcox, Karen E Amaral, Sergio Daniel Marques Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Massachusetts Institute of Technology. Laboratory for Aviation and the Environment Willcox, Karen E Amaral, Sergio Daniel Marques This work proposes an optimization formulation to determine a set of empirical importance weights to achieve a change of probability measure. The objective is to estimate statistics from a target distribution using random samples generated from a (different) proposal distribution. This work considers the specific case in which the proposal distribution from which the random samples are generated is unknown; that is, we have available the samples but no explicit description of their underlying distribution. In this setting, the Radon–Nikodym theorem provides a valid but indeterminable solution to the task, since the distribution from which the random samples are generated is inaccessible. The proposed approach employs the well-defined and determinable empirical distribution function associated with the available samples. The core idea is to compute importance weights associated with the random samples, such that the distance between the weighted proposal empirical distribution function and the desired target distribution function is minimized. The distance metric selected for this work is the L[subscript 2] -norm and the importance weights are constrained to define a probability measure. The resulting optimization problem is shown to be a single linear equality and box-constrained quadratic program. This problem can be solved efficiently using optimization algorithms that scale well to high dimensions. Under some conditions restricting the class of distribution functions, the solution of the optimization problem is shown to result in a weighted proposal empirical distribution function that converges to the target distribution function in the L[subscript 1] -norm, as the number of samples tends to infinity. Results on a variety of test cases show that the proposed approach performs well in comparison with other well-known approaches. Singapore University of Technology and Design. International Design Center United States. Defense Advanced Research Projects Agency (META program through AFRL Contract FA8650-10-C-7083 and Vanderbilt University Contract VUDSR#21807-S7) United States. Federal Aviation Administration. Office of Environment and Energy (FAA Award No. 09-C-NE-MIT, Amendment Nos. 028, 033, and 038) 2017-03-17T19:47:46Z 2017-03-17T19:47:46Z 2016-03 2015-01 2017-02-02T15:21:13Z Article http://purl.org/eprint/type/JournalArticle 0960-3174 1573-1375 http://hdl.handle.net/1721.1/107480 Amaral, Sergio, Douglas Allaire, and Karen Willcox. “Optimal $$L_2$$ L 2 -Norm Empirical Importance Weights for the Change of Probability Measure.” Statistics and Computing (March 14, 2016). https://orcid.org/0000-0003-2156-9338 https://orcid.org/0000-0001-8410-6141 en http://dx.doi.org/10.1007/s11222-016-9644-3 Statistics and Computing Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Allaire, Douglas Willcox, Karen E Amaral, Sergio Daniel Marques Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title | Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title_full | Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title_fullStr | Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title_full_unstemmed | Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title_short | Optimal L[subscript 2]-norm empirical importance weights for the change of probability measure |
| title_sort | optimal l subscript 2 norm empirical importance weights for the change of probability measure |
| url | http://hdl.handle.net/1721.1/107480 https://orcid.org/0000-0003-2156-9338 https://orcid.org/0000-0001-8410-6141 |
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