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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.08122 |
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| _version_ | 1866909196399673344 |
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| author | Derumigny, Alexis Schmidt-Hieber, Johannes |
| author_facet | Derumigny, Alexis Schmidt-Hieber, Johannes |
| contents | We propose a new concept of codivergence, which quantifies the similarity between two probability measures $P_1, P_2$ relative to a reference probability measure $P_0$. In the neighborhood of the reference measure $P_0$, a codivergence behaves like an inner product between the measures $P_1 - P_0$ and $P_2 - P_0$. Codivergences of covariance-type and correlation-type are introduced and studied with a focus on two specific correlation-type codivergences, the $χ^2$-codivergence and the Hellinger codivergence. We derive explicit expressions for several common parametric families of probability distributions. For a codivergence, we introduce moreover the divergence matrix as an analogue of the Gram matrix. It is shown that the $χ^2$-divergence matrix satisfies a data-processing inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08122 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Codivergences and information matrices Derumigny, Alexis Schmidt-Hieber, Johannes Statistics Theory Information Theory Probability 62B11, 46E27, 15A63 We propose a new concept of codivergence, which quantifies the similarity between two probability measures $P_1, P_2$ relative to a reference probability measure $P_0$. In the neighborhood of the reference measure $P_0$, a codivergence behaves like an inner product between the measures $P_1 - P_0$ and $P_2 - P_0$. Codivergences of covariance-type and correlation-type are introduced and studied with a focus on two specific correlation-type codivergences, the $χ^2$-codivergence and the Hellinger codivergence. We derive explicit expressions for several common parametric families of probability distributions. For a codivergence, we introduce moreover the divergence matrix as an analogue of the Gram matrix. It is shown that the $χ^2$-divergence matrix satisfies a data-processing inequality. |
| title | Codivergences and information matrices |
| topic | Statistics Theory Information Theory Probability 62B11, 46E27, 15A63 |
| url | https://arxiv.org/abs/2303.08122 |