Sets of probability measures and convex combination spaces

Miriam Alonso de la Fuente, Pedro Terán
Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:3-10, 2023.

Abstract

The Wasserstein distances between probability distributions are an important tool in modern probability theory which has been generalized to sets of probability distributions. We will show that the (generalized) $L^1$-Wasserstein metric, with the operations of convolution and rescaling, fits in the abstract framework of convex combination spaces: nonlinear metric spaces preserving some of the nice properties of a normed space but accomodating other unusual behaviours. For instance, unlike in a linear space, a singleton $\{P\}$ is typically not convex (it is so only if $P$ is degenerate). Also, some theorems for convex combination spaces are applied to this setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v215-fuente23a, title = {Sets of probability measures and convex combination spaces}, author = {Alonso de la Fuente, Miriam and Ter\'an, Pedro}, booktitle = {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {3--10}, year = {2023}, editor = {Miranda, Enrique and Montes, Ignacio and Quaeghebeur, Erik and Vantaggi, Barbara}, volume = {215}, series = {Proceedings of Machine Learning Research}, month = {11--14 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v215/fuente23a/fuente23a.pdf}, url = {https://proceedings.mlr.press/v215/fuente23a.html}, abstract = {The Wasserstein distances between probability distributions are an important tool in modern probability theory which has been generalized to sets of probability distributions. We will show that the (generalized) $L^1$-Wasserstein metric, with the operations of convolution and rescaling, fits in the abstract framework of convex combination spaces: nonlinear metric spaces preserving some of the nice properties of a normed space but accomodating other unusual behaviours. For instance, unlike in a linear space, a singleton $\{P\}$ is typically not convex (it is so only if $P$ is degenerate). Also, some theorems for convex combination spaces are applied to this setting.} }
Endnote
%0 Conference Paper %T Sets of probability measures and convex combination spaces %A Miriam Alonso de la Fuente %A Pedro Terán %B Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2023 %E Enrique Miranda %E Ignacio Montes %E Erik Quaeghebeur %E Barbara Vantaggi %F pmlr-v215-fuente23a %I PMLR %P 3--10 %U https://proceedings.mlr.press/v215/fuente23a.html %V 215 %X The Wasserstein distances between probability distributions are an important tool in modern probability theory which has been generalized to sets of probability distributions. We will show that the (generalized) $L^1$-Wasserstein metric, with the operations of convolution and rescaling, fits in the abstract framework of convex combination spaces: nonlinear metric spaces preserving some of the nice properties of a normed space but accomodating other unusual behaviours. For instance, unlike in a linear space, a singleton $\{P\}$ is typically not convex (it is so only if $P$ is degenerate). Also, some theorems for convex combination spaces are applied to this setting.
APA
Alonso de la Fuente, M. & Terán, P.. (2023). Sets of probability measures and convex combination spaces. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:3-10 Available from https://proceedings.mlr.press/v215/fuente23a.html.

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