## 2019 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Bernstein's socks and polynomial-time provable coherence Technical Report 2019. Abstract | Links | BibTeX | Tags: bounded rationality, desirability, Quantum mechanics @techreport{Benavoli2019d, title = {Bernstein's socks and polynomial-time provable coherence}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {https://arxiv.org/abs/1903.04406}, year = {2019}, date = {2019-03-12}, abstract = {We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins. }, keywords = {bounded rationality, desirability, Quantum mechanics}, pubstate = {published}, tppubtype = {techreport} } We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins. |

## 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

bayesian statistics biologically inspired bounded rationality Control desirability dual probabilistic programming filtering Gaussian processes lexicographic machine learning Quantum mechanics radar tracking rationality set of probabilities SOS Sum-of-squares polynomials

## 2019 |

Bernstein's socks and polynomial-time provable coherence Technical Report 2019. |

## 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. |