## 2021 |

Casanova, Arianna ; Benavoli, Alessio ; Zaffalon, Marco Nonlinear Desirability as a Linear Classification Problem Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. Abstract | Links | BibTeX | Tags: desirability, imprecise probability @inproceedings{Casanova2021, title = {Nonlinear Desirability as a Linear Classification Problem}, author = {Casanova, Arianna and Benavoli, Alessio and Zaffalon, Marco}, url = {http://alessiobenavoli.com/wp-content/uploads/2021/07/casanova21-1.pdf}, year = {2021}, date = {2021-07-07}, booktitle = {ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR}, abstract = {The present paper proposes a generalization of linearity axioms of coherence through a geometrical approach, which leads to an alternative interpretation of desirability as a classification problem. In particular,we analyze different sets of rationality axioms and,for each one of them, we show that proving that a subject, who provides finite accept and reject statements, respects these axioms, corresponds to solving a binary classification task using, each time, a different (usually nonlinear) family of classifiers. Moreover,by borrowing ideas from machine learning, we show the possibility to define a feature mapping allowing us to reformulate the above nonlinear classification problems as linear ones in a higher-dimensional space.This allows us to interpret gambles directly as payoffs vectors of monetary lotteries, as well as to reduce the task of proving the rationality of a subject to a linear classification task}, keywords = {desirability, imprecise probability}, pubstate = {published}, tppubtype = {inproceedings} } The present paper proposes a generalization of linearity axioms of coherence through a geometrical approach, which leads to an alternative interpretation of desirability as a classification problem. In particular,we analyze different sets of rationality axioms and,for each one of them, we show that proving that a subject, who provides finite accept and reject statements, respects these axioms, corresponds to solving a binary classification task using, each time, a different (usually nonlinear) family of classifiers. Moreover,by borrowing ideas from machine learning, we show the possibility to define a feature mapping allowing us to reformulate the above nonlinear classification problems as linear ones in a higher-dimensional space.This allows us to interpret gambles directly as payoffs vectors of monetary lotteries, as well as to reduce the task of proving the rationality of a subject to a linear classification task |

Benavoli, Alessio; Facchini, Alessandro; Zaffalon, Marco Quantum indistinguishability through exchangeable desirable gambles Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. Abstract | Links | BibTeX | Tags: desirability, Quantum mechanics @inproceedings{benavoli2021quantum, title = {Quantum indistinguishability through exchangeable desirable gambles}, author = {Alessio Benavoli and Alessandro Facchini and Marco Zaffalon}, url = {https://arxiv.org/abs/2105.04336}, year = {2021}, date = {2021-06-01}, booktitle = {ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR}, abstract = {Two particles are identical if all their intrinsic properties, such as spin and charge, are the same, meaning that no quantum experiment can distinguish them. In addition to the well known principles of quantum mechanics, understanding systems of identical particles requires a new postulate, the so called symmetrization postulate. In this work, we show that the postulate corresponds to exchangeability assessments for sets of observables (gambles) in a quantum experiment, when quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles. Finally, we show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems. }, keywords = {desirability, Quantum mechanics}, pubstate = {published}, tppubtype = {inproceedings} } Two particles are identical if all their intrinsic properties, such as spin and charge, are the same, meaning that no quantum experiment can distinguish them. In addition to the well known principles of quantum mechanics, understanding systems of identical particles requires a new postulate, the so called symmetrization postulate. In this work, we show that the postulate corresponds to exchangeability assessments for sets of observables (gambles) in a quantum experiment, when quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles. Finally, we show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems. |

## 2017 |

Benavoli, Alessio ; Facchini, Alessandro ; Vicente-Perez, Jose ; Zaffalon, Marco A polarity theory for sets of desirable gambles Conference Proc. ISIPTA'17 Int. Symposium on Imprecise Probability: Theories and Applications, , PJMLR, 2017. Abstract | Links | BibTeX | Tags: desirability, lexicographic @conference{Benavoli2017c, title = {A polarity theory for sets of desirable gambles}, author = {Benavoli, Alessio and Facchini, Alessandro and Vicente-Perez, Jose and Zaffalon, Marco}, url = {https://arxiv.org/abs/1705.09574}, year = {2017}, date = {2017-05-07}, booktitle = {Proc. ISIPTA'17 Int. Symposium on Imprecise Probability: Theories and Applications, }, pages = {1-12}, publisher = {PJMLR}, abstract = {Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities.}, keywords = {desirability, lexicographic}, pubstate = {published}, tppubtype = {conference} } Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities. |

Benavoli, Alessio ; Colacito, Almudean ; Facchini, Alessandro ; Zaffalon, Marco Accepting and Rejecting Gambles: A Logical Point of View Conference Progic 2017: The 8th Workshop on Combining Probability and Logic, Munich 2017, 2017. Abstract | BibTeX | Tags: desirability @conference{Benavoli2017, title = {Accepting and Rejecting Gambles: A Logical Point of View}, author = { Benavoli, Alessio and Colacito, Almudean and Facchini, Alessandro and Zaffalon, Marco }, year = {2017}, date = {2017-03-30}, booktitle = {Progic 2017: The 8th Workshop on Combining Probability and Logic, Munich 2017}, abstract = {A powerful theory of uncertainty is that of coherent sets of desirable gambles (or desirability). It encompasses, in a uniform way, the Bayesian theory of probability as well as Bayesian robustness and many other theories of uncertainty. In recent years, several attempts have been carried out to explicitly formulate desirability as a logical system; for our purposes, the one by Gillett, Scherl and Shafer is particularly relevant. The goal of this paper is first to provide an appropriate semantics, with the aim to obtain a full completeness result for the logical system of Gillett, Scherl and Shafer. The second goal is to study, from a logical point of view, a generalisation of desirability that allows rejecting gambles to be possible too. Thus we enrich the system by adding a rejection operator, and we formulate some additional deductive rules concerning this operation. The obtained systems are investigated, both from a syntactical and a semantical point of view.}, keywords = {desirability}, pubstate = {published}, tppubtype = {conference} } A powerful theory of uncertainty is that of coherent sets of desirable gambles (or desirability). It encompasses, in a uniform way, the Bayesian theory of probability as well as Bayesian robustness and many other theories of uncertainty. In recent years, several attempts have been carried out to explicitly formulate desirability as a logical system; for our purposes, the one by Gillett, Scherl and Shafer is particularly relevant. The goal of this paper is first to provide an appropriate semantics, with the aim to obtain a full completeness result for the logical system of Gillett, Scherl and Shafer. The second goal is to study, from a logical point of view, a generalisation of desirability that allows rejecting gambles to be possible too. Thus we enrich the system by adding a rejection operator, and we formulate some additional deductive rules concerning this operation. The obtained systems are investigated, both from a syntactical and a semantical point of view. |

# Publications

## 2021 |

Nonlinear Desirability as a Linear Classification Problem Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. |

Quantum indistinguishability through exchangeable desirable gambles Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. |

## 2017 |

A polarity theory for sets of desirable gambles Conference Proc. ISIPTA'17 Int. Symposium on Imprecise Probability: Theories and Applications, , PJMLR, 2017. |

Accepting and Rejecting Gambles: A Logical Point of View Conference Progic 2017: The 8th Workshop on Combining Probability and Logic, Munich 2017, 2017. |