Exact inference on Gaussian graphical models of arbitrary topology using path-sums
We present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from t...
| Main Authors: | , , , |
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| Format: | Journal article |
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Journal of Machine Learning Research
2016
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| _version_ | 1826285400278171648 |
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| author | Giscard, P Choo, Z Thwaite, S Jaksch, D |
| author_facet | Giscard, P Choo, Z Thwaite, S Jaksch, D |
| author_sort | Giscard, P |
| collection | OXFORD |
| description | We present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from the closed-form resummation of infinite families of terms of the walk-sum representation of the covariance matrix. We prove that the path-sum formulation always exists for models whose covariance matrix is positive definite: i.e. it is valid for both walk-summable and non-walk-summable graphical models of arbitrary topology. We show that for graphical models on trees the path-sum formulation is equivalent to Gaussian belief propagation. We also recover, as a corollary, an existing result that uses determinants to calculate the covariance matrix. We show that the path-sum formulation formulation is valid for arbitrary partitions of the inverse covariance matrix. We give detailed examples demonstrating our results. |
| first_indexed | 2024-03-07T01:28:15Z |
| format | Journal article |
| id | oxford-uuid:92b4fffc-cac0-473a-b52b-9a15e8efbbbb |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:28:15Z |
| publishDate | 2016 |
| publisher | Journal of Machine Learning Research |
| record_format | dspace |
| spelling | oxford-uuid:92b4fffc-cac0-473a-b52b-9a15e8efbbbb2022-03-26T23:27:26ZExact inference on Gaussian graphical models of arbitrary topology using path-sumsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:92b4fffc-cac0-473a-b52b-9a15e8efbbbbSymplectic Elements at OxfordJournal of Machine Learning Research2016Giscard, PChoo, ZThwaite, SJaksch, DWe present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from the closed-form resummation of infinite families of terms of the walk-sum representation of the covariance matrix. We prove that the path-sum formulation always exists for models whose covariance matrix is positive definite: i.e. it is valid for both walk-summable and non-walk-summable graphical models of arbitrary topology. We show that for graphical models on trees the path-sum formulation is equivalent to Gaussian belief propagation. We also recover, as a corollary, an existing result that uses determinants to calculate the covariance matrix. We show that the path-sum formulation formulation is valid for arbitrary partitions of the inverse covariance matrix. We give detailed examples demonstrating our results. |
| spellingShingle | Giscard, P Choo, Z Thwaite, S Jaksch, D Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title | Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title_full | Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title_fullStr | Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title_full_unstemmed | Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title_short | Exact inference on Gaussian graphical models of arbitrary topology using path-sums |
| title_sort | exact inference on gaussian graphical models of arbitrary topology using path sums |
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