Pinsker's inequality for adapted total variation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911026175279104 |
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| author | Beiglböck, Mathias Zona, Markus |
| author_facet | Beiglböck, Mathias Zona, Markus |
| contents | Pinsker's classical inequality asserts that the total variation $TV(μ, ν)$ between two probability measures is bounded by $\sqrt{ 2H(μ|ν)}$ where $H$ denotes the relative entropy (or Kullback-Leibler divergence). Considering the discrete metric, $TV$ can be seen as a Wasserstein distance and as such possesses an adapted variant $ATV$. Adapted Wasserstein distances have distinct advantages over their classical counterparts when $μ, ν$ are the laws of stochastic processes $(X_k)_{k=1}^n, (Y_k)_{k=1}^n$ and exhibit numerous applications from stochastic control to machine learning. In this note we observe that the adapted total variation distance $ATV$ satisfies the Pinsker-type inequality $$ ATV(μ, ν)\leq \sqrt{n} \sqrt{2 H(μ|ν)}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pinsker's inequality for adapted total variation Beiglböck, Mathias Zona, Markus Probability Information Theory Pinsker's classical inequality asserts that the total variation $TV(μ, ν)$ between two probability measures is bounded by $\sqrt{ 2H(μ|ν)}$ where $H$ denotes the relative entropy (or Kullback-Leibler divergence). Considering the discrete metric, $TV$ can be seen as a Wasserstein distance and as such possesses an adapted variant $ATV$. Adapted Wasserstein distances have distinct advantages over their classical counterparts when $μ, ν$ are the laws of stochastic processes $(X_k)_{k=1}^n, (Y_k)_{k=1}^n$ and exhibit numerous applications from stochastic control to machine learning. In this note we observe that the adapted total variation distance $ATV$ satisfies the Pinsker-type inequality $$ ATV(μ, ν)\leq \sqrt{n} \sqrt{2 H(μ|ν)}.$$ |
| title | Pinsker's inequality for adapted total variation |
| topic | Probability Information Theory |
| url | https://arxiv.org/abs/2506.22106 |