International Journal of Approximate Reasoning, 2015
This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 8... more This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 8th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA'13). The symposium was organized by the Society for Imprecise Probability: Theories and Applications (SIPTA) at the Université de Technologie de Compiègne (France) in July 2013 (http :/ /www.sipta .org /isipta13). The biennial ISIPTA meetings are well established among international conferences on generalized methods for uncertainty quantification. The first ISIPTA took place in Gent in 1999, followed by meetings in Cornell, Lugano, Carnegie Mellon, Prague, Durham and Innsbruck. Compiègne proved to be a very nice location for ISIPTA 2013, offering wonderful opportunities for collaborations and discussions, as well as sightseeing places such as its imperial palace. Following a selective refereeing process, 38 papers were selected for ISIPTA'13 and presented in plenary sessions followed by poster sessions enabling deeper discussions. Following previous years tradition, eight poster-only presentations were added to those papers. These presentations were complemented by tutorials given by Matthias C.M. Troffaes and Thierry Denoeux on imprecise probability theory and belief functions, respectively. A number of invited talks about topics related to imprecise probabilities were also given: Alessio Benavoli gave a talk on "Pushing Dynamic Estimation to the Extremes: from the Moon to Imprecise Probability", Isaac Elishakoff on the "Recent Developments in Applied Mechanics with Uncertainties", Christophe Labreuche on "Robustness in Multi-Criteria Decision Making and its relation with Imprecise Probabilities" and Jean-Marc Tallon on "Ambiguity and Ambiguity Attitudes in Economics". The biennial IJAR Young Researcher Award, generously provided by Elsevier, was awarded at the meeting. The Prize in Gold was awarded to Andrea Wiencierz (Germany) and Ignacio Montes Gutiérrez (Spain), while Rocco de Rosa (Italy) received an honorable mention.
Credal sets containing coherent conditional probabilities defined by Hausdorff measures on the Bo... more Credal sets containing coherent conditional probabilities defined by Hausdorff measures on the Borel sigmafield of metric spaces with bi-Lipschitz equivalent metrics, are proven to represent merging opinions with increasing information.
In this paper, we explore the use of aggregation functions in the construction of coherent upper ... more In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [0,∞[n is presented. Lastly, some concluding remarks are added.
A new model of coherent upper conditional previsions is proposed to represent uncertainty and to ... more A new model of coherent upper conditional previsions is proposed to represent uncertainty and to make previsions in complex systems. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension, it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below.Moreover sufficient conditions are given such that the upper conditional previsions satisfy the disintegration property and the conglomerability principle.
In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures... more In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures on the linear space of all absolutely Choquet integrable random variables. They are proven to satisfy the disintegration property on every non-null partition if Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Coherent upper conditional probabilities are obtained as restrictions to the indicators function and it is proven they can be extended as finitely additive probabilities in the sense of Dubins. Examples are given in the discrete metric space and in the Euclidean metric space.
Formulating diagnosis is a complex process, related to the clinician’s ability to represent the p... more Formulating diagnosis is a complex process, related to the clinician’s ability to represent the patient’s discomfort, to use error due to the incompleteness of the information available, to make predictions about wellbeing. We interpret the therapist-patient system as a complex system, whose evolution, representing the phase of alliance, is described by a finite family of contractions that, starting from certain initial conditions, evolve the system into the attractor; this set, characterized by its own complexity, measured in terms of the Hausdor dimension, represents the state in which the therapist and patient find themselves after the phase of alliance. Updating the Level of Knowledge and Making a Successful Diagnosis A probabilistic approach of the diagnostic process is proposed in which the subject’s degree of knowledge is represented with coherent upper conditional probabilities defined by Hausdor outer measures [1]. Using this model, the diagnosis is assumed to be positive w...
Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational... more Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational Reasoning produce different preference ordering between alternatives. From a mathematical point of view, complex decisions can be defined as decisions where a preference ordering between random variables cannot be represented by a linear functional. The Asymmetric and the Symmetric Choquet integrals with respect to non additive-measures have been defined as aggregation operators of data sets and as a tool to assess an ordering between random variables. They could be considered to represent preference orderings of the conscious and unconscious mind when a human being make decision. Sufficient conditions are given such that the two integral representations of a coherent upper conditional prevision by the Asymmetric Choquet integral and the Symmetric Choquet integral with respect to Hausdorff outer measures coincide and linearity holds.
In questi ultimi anni il ccn'ello c la sua attivita‘ sono al centro degli studi di scienziati... more In questi ultimi anni il ccn'ello c la sua attivita‘ sono al centro degli studi di scienziati e ricercatori di diverse discipline. Anche i matematici sono intenti nello studio di modelli che simulino l'attivita' cerebrale e aiutino a capire, almeno in parte i meccanismi che la regolano. Sicuramente tra le capacita‘ piu' interessanti del cervello c’e’ quella di apprendere, di imparare cioe' dagli errori e dall'esperienza. Non secondaria e' poi la capacita' di memorizzare un'immagine, un odore o un suono e riconoscerlo sotto particolari stimoli. Nessun computer. finora esistente, riesce a memorizzare un numero di informazioni prossimo a quello che e' capace di contenere il cervello c soprattutto a gestire tutti i dati a sua disposizione in tempi brevissimi. In questo lavoro viene presentato il modello di Hopfield. come esempio di schematizzazione dell'attivita' neuronale. con particolare riferimento al processo di memorizzazione. L'i...
Modeling Decisions for Artificial Intelligence, 2020
In this paper, the model of coherent upper and lower conditional previsions is proposed to repres... more In this paper, the model of coherent upper and lower conditional previsions is proposed to represent preference orderings and equivalences between random variables that human beings consider consciously or unconsciously when making decisions. To solve the contradiction, Linda’s Problem (i.e., conjunction fallacy) is re-interpreted in terms of the probabilistic model based on coherent upper and lower conditional probabilities. Main insights of this mathematical solution for modeling decisions in AI are evidenced accordingly.
Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed... more Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed to define coherent lower previsions on the class of all bounded functions. It is the extension to the class of all bounded random variables of a shift-invariant collection integral with respect to a collection D and a capacity μ. Related coherent upper previsions are also considered.
Preference orderings assigned by coherent lower and upper conditional previsions are defined and ... more Preference orderings assigned by coherent lower and upper conditional previsions are defined and they are considered to define maximal random variables and Bayes random variables. Sufficient conditions are given such that a random variable is maximal if and only if it is a Bayes random variable. In a metric space preference orderings represented by coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures are given.
Motivations Ammonites are extinct ectococled molluscs belonging to the Class Cephalopoda which li... more Motivations Ammonites are extinct ectococled molluscs belonging to the Class Cephalopoda which lived during the Mesozoic Era. Their usefulness in Jurassic and Cretaceous paleontology and biostratigraphy study is widely proved. For this reason, they are studied by several authors worldwide in order to achieve information regarding their habitats and climate of past world. Coherent upper conditional previsions defined with respect to Hausdorff outer measures are used to make a probabilistic analysis of the paleoenvironmental causes that generated complex sutural lines. In particular, the role of hydrostatic pressure is studied.
Coherent lower previsions generalize the expected values and they are defined on the class of all... more Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-ad...
Annals of Mathematics and Artificial Intelligence, 2021
The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer ... more The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer measures, is proposed to represent the preference orderings and the equivalences, respectively assigned by the conscious and unconscious thought in human decision making under uncertainty. Complexity of partial information is represented by the Hausdorff dimension of the conditioning event. When the events, that describe the decision problem, are measurable is represented to the s-dimensional Hausdorff outer measure, where s is the Hausdorff dimension of the conditioning event, an optimal decision can be reached. The model is applied and discussed in Linda’s Problem and the conjunction fallacy is resolved.
Let (Ω, d) be a metric space and let B be a partition of Ω. For every B in B let L(B) be the clas... more Let (Ω, d) be a metric space and let B be a partition of Ω. For every B in B let L(B) be the class of all bounded random variables on B. Upper conditional previsions P (X|B) are defined on L(B) × B with respect to a class of Hausdorff outer measures when the conditioning event B has positive and finite Hausdorff outer measure in its dimension; otherwise they are defined by a 0-1 valued finitely additive (but not countably additive) probability. For every conditioning event B these upper conditional previsions P (X|B) are proven to be the upper envelopes of all linear previsions, defined on the class of all bounded random variables on B and dominated by P (X|B). Upper conditional probabilities are obtained as a particular case when L(B) is the class of all 0-1 valued random variables on B. The unconditional upper probability is defined when the conditioning event is Ω. Relations among different types of convergence of sequences of random variables are investigated with respect to the...
International Journal of Approximate Reasoning, 2015
This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 8... more This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 8th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA'13). The symposium was organized by the Society for Imprecise Probability: Theories and Applications (SIPTA) at the Université de Technologie de Compiègne (France) in July 2013 (http :/ /www.sipta .org /isipta13). The biennial ISIPTA meetings are well established among international conferences on generalized methods for uncertainty quantification. The first ISIPTA took place in Gent in 1999, followed by meetings in Cornell, Lugano, Carnegie Mellon, Prague, Durham and Innsbruck. Compiègne proved to be a very nice location for ISIPTA 2013, offering wonderful opportunities for collaborations and discussions, as well as sightseeing places such as its imperial palace. Following a selective refereeing process, 38 papers were selected for ISIPTA'13 and presented in plenary sessions followed by poster sessions enabling deeper discussions. Following previous years tradition, eight poster-only presentations were added to those papers. These presentations were complemented by tutorials given by Matthias C.M. Troffaes and Thierry Denoeux on imprecise probability theory and belief functions, respectively. A number of invited talks about topics related to imprecise probabilities were also given: Alessio Benavoli gave a talk on "Pushing Dynamic Estimation to the Extremes: from the Moon to Imprecise Probability", Isaac Elishakoff on the "Recent Developments in Applied Mechanics with Uncertainties", Christophe Labreuche on "Robustness in Multi-Criteria Decision Making and its relation with Imprecise Probabilities" and Jean-Marc Tallon on "Ambiguity and Ambiguity Attitudes in Economics". The biennial IJAR Young Researcher Award, generously provided by Elsevier, was awarded at the meeting. The Prize in Gold was awarded to Andrea Wiencierz (Germany) and Ignacio Montes Gutiérrez (Spain), while Rocco de Rosa (Italy) received an honorable mention.
Credal sets containing coherent conditional probabilities defined by Hausdorff measures on the Bo... more Credal sets containing coherent conditional probabilities defined by Hausdorff measures on the Borel sigmafield of metric spaces with bi-Lipschitz equivalent metrics, are proven to represent merging opinions with increasing information.
In this paper, we explore the use of aggregation functions in the construction of coherent upper ... more In this paper, we explore the use of aggregation functions in the construction of coherent upper previsions. Sub-additivity is one of the defining properties of a coherent upper prevision defined on a linear space of random variables and thus we introduce a new sub-additive transformation of aggregation functions, called a revenue transformation, whose output is a sub-additive aggregation function bounded below by the transformed aggregation function, if it exists. Method of constructing coherent upper previsions by means of shift-invariant, positively homogeneous and sub-additive aggregation functions is given and a full characterization of shift-invariant, positively homogeneous and idempotent aggregation functions on [0,∞[n is presented. Lastly, some concluding remarks are added.
A new model of coherent upper conditional previsions is proposed to represent uncertainty and to ... more A new model of coherent upper conditional previsions is proposed to represent uncertainty and to make previsions in complex systems. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension, it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below.Moreover sufficient conditions are given such that the upper conditional previsions satisfy the disintegration property and the conglomerability principle.
In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures... more In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures on the linear space of all absolutely Choquet integrable random variables. They are proven to satisfy the disintegration property on every non-null partition if Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Coherent upper conditional probabilities are obtained as restrictions to the indicators function and it is proven they can be extended as finitely additive probabilities in the sense of Dubins. Examples are given in the discrete metric space and in the Euclidean metric space.
Formulating diagnosis is a complex process, related to the clinician’s ability to represent the p... more Formulating diagnosis is a complex process, related to the clinician’s ability to represent the patient’s discomfort, to use error due to the incompleteness of the information available, to make predictions about wellbeing. We interpret the therapist-patient system as a complex system, whose evolution, representing the phase of alliance, is described by a finite family of contractions that, starting from certain initial conditions, evolve the system into the attractor; this set, characterized by its own complexity, measured in terms of the Hausdor dimension, represents the state in which the therapist and patient find themselves after the phase of alliance. Updating the Level of Knowledge and Making a Successful Diagnosis A probabilistic approach of the diagnostic process is proposed in which the subject’s degree of knowledge is represented with coherent upper conditional probabilities defined by Hausdor outer measures [1]. Using this model, the diagnosis is assumed to be positive w...
Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational... more Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational Reasoning produce different preference ordering between alternatives. From a mathematical point of view, complex decisions can be defined as decisions where a preference ordering between random variables cannot be represented by a linear functional. The Asymmetric and the Symmetric Choquet integrals with respect to non additive-measures have been defined as aggregation operators of data sets and as a tool to assess an ordering between random variables. They could be considered to represent preference orderings of the conscious and unconscious mind when a human being make decision. Sufficient conditions are given such that the two integral representations of a coherent upper conditional prevision by the Asymmetric Choquet integral and the Symmetric Choquet integral with respect to Hausdorff outer measures coincide and linearity holds.
In questi ultimi anni il ccn'ello c la sua attivita‘ sono al centro degli studi di scienziati... more In questi ultimi anni il ccn'ello c la sua attivita‘ sono al centro degli studi di scienziati e ricercatori di diverse discipline. Anche i matematici sono intenti nello studio di modelli che simulino l'attivita' cerebrale e aiutino a capire, almeno in parte i meccanismi che la regolano. Sicuramente tra le capacita‘ piu' interessanti del cervello c’e’ quella di apprendere, di imparare cioe' dagli errori e dall'esperienza. Non secondaria e' poi la capacita' di memorizzare un'immagine, un odore o un suono e riconoscerlo sotto particolari stimoli. Nessun computer. finora esistente, riesce a memorizzare un numero di informazioni prossimo a quello che e' capace di contenere il cervello c soprattutto a gestire tutti i dati a sua disposizione in tempi brevissimi. In questo lavoro viene presentato il modello di Hopfield. come esempio di schematizzazione dell'attivita' neuronale. con particolare riferimento al processo di memorizzazione. L'i...
Modeling Decisions for Artificial Intelligence, 2020
In this paper, the model of coherent upper and lower conditional previsions is proposed to repres... more In this paper, the model of coherent upper and lower conditional previsions is proposed to represent preference orderings and equivalences between random variables that human beings consider consciously or unconsciously when making decisions. To solve the contradiction, Linda’s Problem (i.e., conjunction fallacy) is re-interpreted in terms of the probabilistic model based on coherent upper and lower conditional probabilities. Main insights of this mathematical solution for modeling decisions in AI are evidenced accordingly.
Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed... more Given a finite non-empty set Ω a new type of integral, named super-additive integral, is proposed to define coherent lower previsions on the class of all bounded functions. It is the extension to the class of all bounded random variables of a shift-invariant collection integral with respect to a collection D and a capacity μ. Related coherent upper previsions are also considered.
Preference orderings assigned by coherent lower and upper conditional previsions are defined and ... more Preference orderings assigned by coherent lower and upper conditional previsions are defined and they are considered to define maximal random variables and Bayes random variables. Sufficient conditions are given such that a random variable is maximal if and only if it is a Bayes random variable. In a metric space preference orderings represented by coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures are given.
Motivations Ammonites are extinct ectococled molluscs belonging to the Class Cephalopoda which li... more Motivations Ammonites are extinct ectococled molluscs belonging to the Class Cephalopoda which lived during the Mesozoic Era. Their usefulness in Jurassic and Cretaceous paleontology and biostratigraphy study is widely proved. For this reason, they are studied by several authors worldwide in order to achieve information regarding their habitats and climate of past world. Coherent upper conditional previsions defined with respect to Hausdorff outer measures are used to make a probabilistic analysis of the paleoenvironmental causes that generated complex sutural lines. In particular, the role of hydrostatic pressure is studied.
Coherent lower previsions generalize the expected values and they are defined on the class of all... more Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-ad...
Annals of Mathematics and Artificial Intelligence, 2021
The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer ... more The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer measures, is proposed to represent the preference orderings and the equivalences, respectively assigned by the conscious and unconscious thought in human decision making under uncertainty. Complexity of partial information is represented by the Hausdorff dimension of the conditioning event. When the events, that describe the decision problem, are measurable is represented to the s-dimensional Hausdorff outer measure, where s is the Hausdorff dimension of the conditioning event, an optimal decision can be reached. The model is applied and discussed in Linda’s Problem and the conjunction fallacy is resolved.
Let (Ω, d) be a metric space and let B be a partition of Ω. For every B in B let L(B) be the clas... more Let (Ω, d) be a metric space and let B be a partition of Ω. For every B in B let L(B) be the class of all bounded random variables on B. Upper conditional previsions P (X|B) are defined on L(B) × B with respect to a class of Hausdorff outer measures when the conditioning event B has positive and finite Hausdorff outer measure in its dimension; otherwise they are defined by a 0-1 valued finitely additive (but not countably additive) probability. For every conditioning event B these upper conditional previsions P (X|B) are proven to be the upper envelopes of all linear previsions, defined on the class of all bounded random variables on B and dominated by P (X|B). Upper conditional probabilities are obtained as a particular case when L(B) is the class of all 0-1 valued random variables on B. The unconditional upper probability is defined when the conditioning event is Ω. Relations among different types of convergence of sequences of random variables are investigated with respect to the...
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