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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