On the entropy and information of Gaussian mixtures
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916127412584448 |
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| author | Eskenazis, Alexandros Gavalakis, Lampros |
| author_facet | Eskenazis, Alexandros Gavalakis, Lampros |
| contents | We establish several convexity properties for the entropy and Fisher information of mixtures of centered Gaussian distributions. First, we prove that if $X_1, X_2$ are independent scalar Gaussian mixtures, then the entropy of $\sqrt{t}X_1 + \sqrt{1-t}X_2$ is concave in $t \in [0,1]$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $\mathbb{R}^d$. As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15997 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the entropy and information of Gaussian mixtures Eskenazis, Alexandros Gavalakis, Lampros Information Theory Probability 94A17 (Primary) 60E15, 26B25 (Secondary) We establish several convexity properties for the entropy and Fisher information of mixtures of centered Gaussian distributions. First, we prove that if $X_1, X_2$ are independent scalar Gaussian mixtures, then the entropy of $\sqrt{t}X_1 + \sqrt{1-t}X_2$ is concave in $t \in [0,1]$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $\mathbb{R}^d$. As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions. |
| title | On the entropy and information of Gaussian mixtures |
| topic | Information Theory Probability 94A17 (Primary) 60E15, 26B25 (Secondary) |
| url | https://arxiv.org/abs/2308.15997 |