A generalized notion of conjunction for two conditional events

Lydia Castronovo; Giuseppe Sanfilippo

A generalized notion of conjunction for two conditional events

Lydia Castronovo, Giuseppe Sanfilippo

Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:96-108, 2023.

Abstract

Traditionally the conjunction of conditional events has been defined as a three-valued object. However, in this way classical logical and probabilistic properties are not preserved. In recent literature, a notion of conjunction of two conditional events as a suitable conditional random quantity, which satisfies classical probabilistic properties, has been deepened in the setting of coherence. In this framework the conjunction

$(A|H) \wedge (B|K)$ of two conditional events

$A|H$ and

$B|K$ is defined as a five-valued object with set of possible values

$\{1,0,x,y,z\}$ , where

$x=P(A|H), y=P(B|K)$ , and

$z=\mathbb{P}[(A|H) \wedge (B|K)]$ . In this paper we propose a generalization of this object, denoted by

$(A|H) \wedge_{a,b} (B|K)$ , where the values

$x$ and

$y$ are replaced by two arbitrary values

$a,b \in [0,1]$ . Then, by means of a geometrical approach, we compute the set of all coherent assessments on the family

$\{A|H,B|K,(A|H) \wedge_{a,b} (B|K)\}$ , by also showing that in the general case the Fréchet-Hoeffding bounds for the conjunction are not satisfied. We also analyze some particular cases. Finally, we study coherence in the imprecise case of an interval-valued probability assessment and we consider further aspects on

$(A|H) \wedge_{a,b} (B|K)$ .

Cite this Paper

BibTeX


@InProceedings{pmlr-v215-castronovo23a,
  title = 	 {A generalized notion of conjunction for two conditional events},
  author =       {Castronovo, Lydia and Sanfilippo, Giuseppe},
  booktitle = 	 {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications},
  pages = 	 {96--108},
  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/castronovo23a/castronovo23a.pdf},
  url = 	 {https://proceedings.mlr.press/v215/castronovo23a.html},
  abstract = 	 {Traditionally the conjunction of conditional events has been defined as a three-valued object. However, in this way classical logical and probabilistic properties are not preserved. In recent literature, a notion of conjunction of two conditional events as a suitable conditional random quantity, which satisfies classical probabilistic properties, has been deepened in the setting of coherence. In this framework the conjunction $(A|H) \wedge (B|K)$ of two conditional events $A|H$ and $B|K$ is defined as a five-valued object with set of possible values $\{1,0,x,y,z\}$, where $x=P(A|H), y=P(B|K)$, and $z=\mathbb{P}[(A|H) \wedge (B|K)]$. In this paper we propose a generalization of this object, denoted by $(A|H) \wedge_{a,b} (B|K)$, where the values $x$ and $y$ are replaced by two arbitrary values $a,b \in [0,1]$. Then, by means of a geometrical approach, we compute the set of all coherent assessments on the family $\{A|H,B|K,(A|H) \wedge_{a,b} (B|K)\}$, by also showing that in the general case the Fréchet-Hoeffding bounds for the conjunction are not satisfied. We also analyze some particular cases. Finally, we study coherence in the imprecise case of an interval-valued probability assessment and we consider further aspects on $(A|H) \wedge_{a,b} (B|K)$.}
}

Endnote

%0 Conference Paper
%T A generalized notion of conjunction for two conditional events
%A Lydia Castronovo
%A Giuseppe Sanfilippo
%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-castronovo23a
%I PMLR
%P 96--108
%U https://proceedings.mlr.press/v215/castronovo23a.html
%V 215
%X Traditionally the conjunction of conditional events has been defined as a three-valued object. However, in this way classical logical and probabilistic properties are not preserved. In recent literature, a notion of conjunction of two conditional events as a suitable conditional random quantity, which satisfies classical probabilistic properties, has been deepened in the setting of coherence. In this framework the conjunction $(A|H) \wedge (B|K)$ of two conditional events $A|H$ and $B|K$ is defined as a five-valued object with set of possible values $\{1,0,x,y,z\}$, where $x=P(A|H), y=P(B|K)$, and $z=\mathbb{P}[(A|H) \wedge (B|K)]$. In this paper we propose a generalization of this object, denoted by $(A|H) \wedge_{a,b} (B|K)$, where the values $x$ and $y$ are replaced by two arbitrary values $a,b \in [0,1]$. Then, by means of a geometrical approach, we compute the set of all coherent assessments on the family $\{A|H,B|K,(A|H) \wedge_{a,b} (B|K)\}$, by also showing that in the general case the Fréchet-Hoeffding bounds for the conjunction are not satisfied. We also analyze some particular cases. Finally, we study coherence in the imprecise case of an interval-valued probability assessment and we consider further aspects on $(A|H) \wedge_{a,b} (B|K)$.

APA


Castronovo, L. & Sanfilippo, G.. (2023). A generalized notion of conjunction for two conditional events. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:96-108 Available from https://proceedings.mlr.press/v215/castronovo23a.html.

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