Sets of probability measures and convex combination spaces

Miriam Alonso de la Fuente; Pedro Terán

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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