V Laquerbe
email: vincent.laquerbe@isae.fr

R Pascaud

T Callegari

L Liard

O Pascal

-Saint-Malo Antenne VHF miniature reconfigurable en fréquence par décharge plasma : modélisation et résultats expérimentaux

Dans ce papier, nous proposons l'utilisation d'un plasma froid hors équilibre thermodynamique comme matériau à permittivité négative dans la réalisation d'une antenne sphérique miniature à résonance quasi-statique. Cette résonance est étudiée analytiquement puis numériquement et un prototype expérimental est présenté.

Introduction

La miniaturisation d'antenne est un challenge important dans le design des systèmes actuels et en particulier dans la bande de fréquences VHF où les longueurs d'onde sont importantes. De nombreuses techniques de miniaturisation ont été proposées pour ces fréquences [START_REF] Volakis | Small Antennas Miniaturization Techniques & Applications[END_REF]. Récemment, l'utilisation d'un milieu à permittivité négative (ENG pour Epsilon NeGative) a été suggérée afin de rendre résonantes des structures sub-longueur d'onde [START_REF] Ziolkowski | Metamaterial-based efficient electrically small antennas[END_REF][START_REF] Stuart | Electrically small antenna elements using negative permittivity resonators[END_REF].

Le principe théorique repose sur le comportement électromagnétique d'une petite sphère (au sens de la longueur d'onde). Sous hypothèse quasi-statique, sa polarisabilité α, soit sa capacité à rétro-diffuser la puissance qu'elle reçoit, dépend de sa constante diélectrique ε p et s'écrit [START_REF] Stuart | Electrically small antenna elements using negative permittivity resonators[END_REF] :

α = 3 ε p -1 ε p + 2 (1)
On remarque, d'après l'équation [START_REF] Volakis | Small Antennas Miniaturization Techniques & Applications[END_REF], que cette polarisabilité tend vers l'infini lorsque ε p = -2. C'est ce comportement particulier qui a été envisagé pour la conception d'antenne miniature [START_REF] Stuart | Electrically small antenna elements using negative permittivity resonators[END_REF].

L'implémentation de cette solution a par exemple été réalisée avec des métamatériaux [START_REF] Smith | Loop-wire medium for investigating plasmons at microwave frequencies[END_REF]. Nous proposons ici d'utiliser la réponse diélectrique naturelle d'un plasma non-magnétisé froid et hors équilibre thermodynamique pour obtenir un milieu dont la permittivité est négative. Ces plasmas sont en effet représentés du point de vue électromagnétique par leur permittivité relative complexe qui suit le modèle de Drude [START_REF] Lieberman | Principles of Plasma Discharges and Materials Processing[END_REF] :

ε p (ω) = 1 - ω 2 p ω(ω + iν) , ω 2 p = e 2 n e m e ε 0 (2)
où ν, ω p , e, n e , m e et ε 0 sont respectivement la fréquence de collision (pertes), la fréquence angulaire du plasma, la charge électrique élémentaire, la densité électronique, la masse de l'électron et la permittivité du vide. Le contrôle de la densité électronique du plasma permet par ailleurs de modifier sa constante diélectrique dans un gamme relativement étendue (valeurs négatives, nulles ou comprises entre 0 et 1) et ainsi de rendre l'antenne potentiellement reconfigurable en fréquence. Toutefois, l'hétérogénéité du plasma requiert une étude analytique plus poussée.

Dans ce papier, nous présentons des études analytiques, numériques et expérimentales d'un prototype antennaire à base de plasma dont les différentes étapes sont présentées sur la Figure 1. Dans un premier temps, un profil de densité électronique radial est mesuré. Ce profil crée, à une fréquence donnée, un milieu dont on connait la permittivité à l'aide de l'équation [START_REF] Ziolkowski | Metamaterial-based efficient electrically small antennas[END_REF], que l'on exprime sous une forme polynomiale en r. On résout ensuite analytiquement l'équation de Laplace pour différentes fréquences et on en déduit l'existence de résonances (forte amplification de la polarisabilité de la structure). Par ailleurs, en discrétisant suffisamment le profil mesuré précédemment en calottes sphérique de pas homogène, on est en mesure d'étudier le résonateur ENG grâce à des logiciels de simulation électromagnétiques 3D.

Description du prototype

La Figure 2 présente le prototype développé dans le cadre de ces travaux. Une hémisphère de verre et un disque de verre sont placés de part et d'autre d'un support métallique, faisant office de plan de masse pour l'antenne, afin de créer un cavité hermétique dans laquelle est excité un plasma. Le disque de verre et le support sont percés en leur centre pour permettre le passage d'un connecteur traversant de type SMA sur lequel est ensuite vissé un monopole.

Afin de créer une structure ENG résonante, une décharge plasma est générée dans l'enceinte hémisphérique. Une solution technique originale a ainsi été implémenté pour ne pas perturber le fonctionnement de l'antenne monopole. Pour cela, une source de couplage par induction (ICP pour Inductively Coupled Plasma) a été placée sous le plan de masse. Celle-ci est composée d'un générateur de puissance à 13.56 MHz en série avec une spirale plane de 145 mm de diamètre extérieur qui comprend un enroulement de pas homogène sur 3 tours et demi. Le couplage entre la spirale et l'enceinte est rendu possible par l'usinage de six fentes radiales dans le plan de masse (voir Figure 2). Des tiges métalliques radiales supplémentaires ont également été ajoutées sur ces fentes ; ce qui permet de réduire la diffraction du monopole sur la partie inférieure et de filtrer la composante radiale du champ électrique de la source ICP. Cet écran de Faraday augmente le transfert de puissance entre la spirale et la décharge plasma en diminuant la composante couplée capacitivement.

Mesures de densités électroniques

Afin d'effectuer des mesures de densité, l'hémisphère supérieure a été légèrement modifiée et comprend plusieurs accès DN16. Une sonde de Langmuir [START_REF] Mott-Smith | The theory of collectors in gaseous discharges[END_REF], montée sur un translateur linéaire (voir Figure 2), est ensuite

Modélisation analytique du résonateur

Comme on peut le voir sur la Figure 4, la densité électronique n'est pas constante dans l'enceinte, du fait de la diffusion des particules, et sa valeur au bord de la sphère chute rapidement. Cet effet est principalement dû à l'apparition d'une gaine (région électriquement neutre) nécessairement présente entre le gaz ionisé et le milieu environnant, ici le verre. Afin de prendre en compte cette hétérogénéité, le profil de densité électronique mesuré n e (r) est interpolé par une fonction polynomiale sur l'intervalle [0, r p ], où r p représente le rayon de l'hémisphère dans laquelle la décharge plasma s'établit. Plusieurs polynômes de degrés différents ont été considérés. Le polynôme de degré 3 de la Figure 4 (courbe bleue) est cohérent avec la mesure. A l'aide de l'équation (2), on en déduit, pour chaque fréquence de fonctionnement, le profil de permittivité relative suivant :

ε p = ε p,0 + ε p,1 r + ε p,2 r 2 + ε p,3 r 3 (3)
De r p à r s (rayon intérieur de l'hémisphère de verre) s'établit la gaine que l'on assimile à un milieu très faiblement ionisé dont la permittivité relative ε s est supposée égale à 1. On suppose d'après la Figure 4 que l'épaisseur de la gaine r s -r p est égale à 2 mm. L'hémisphère de verre est représenté par une calotte de rayon intérieur r s = 102 mm et de rayon extérieur r d = 107.5 mm. Sa permittivité relative est ε d = 5.5.

On s'intéresse maintenant au problème analytique complet présenté sur la Figure 5, ce qui se justifie en invoquant la théorie des images sur la structure réelle avec plan de masse. Pour cela, on utilise le profil mesuré précédemment que l'on extrapole par symétrie centrale de manière à reconstituer un milieu sphérique dont les paramètres diélectriques varient uniquement avec la position radiale. On discutera plus loin de la validité de cette extrapolation. La polarisabilité de cette structure se calcule analytiquement en résolvant l'équation de Laplace pour le profil continu de l'équation [START_REF] Stuart | Electrically small antenna elements using negative permittivity resonators[END_REF]. L'indépendance angulaire de la géométrie simplifie le problème au seul calcul des composantes radiales dont l'expression générale est donnée par l'équation (4) [START_REF] Sihvola | Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers[END_REF].

∂ ∂r r 2 ε p (r) ∂R(r) ∂r -2ε p (r)R(r) = 0 (4)
A l'intérieur de la gaine, du verre et pour le milieu extérieur, la permittivité relative est uniforme et donc seule la composante dipolaire est non nulle. En revanche, à l'intérieur du plasma, la composante radiale s'exprime selon une série de r dont les termes sont calculés par récurrence selon l'analyse présentée dans [START_REF] Laquerbe | Analytical model for the study of quasistatic resonance of non-uniform ε-negative spheres[END_REF]. On retrouve l'expression finale du potentiel scalaire en injectant le premier terme de cette série dans les conditions aux limites à chaque interface entre deux milieux contigus, comme suggéré dans [START_REF] Sihvola | Transmission line analogy for calculating the effective permittivity of mixtures with spherical multilayer scatterers[END_REF].

On calcule maintenant le coefficient dipolaire B OU T (normalisé par λ 2 ) de l'expression du potentiel scalaire dans le milieu extérieur pour différentes fréquences. La Figure 6 présente l'évolution de l'amplitude de B OU T en fonction de la fréquence. Dans la bande de fréquence considérée, pour laquelle le facteur de miniaturisation ka < 0.5, et pour le profil de densité radiale mesuré de la Figure 4,on n'observe pas d'amplification significative de B OU T . A première vue, la résonance est attendue à plus haute fréquence en raison des valeurs élevées de densité obtenues.

Néanmoins, la résonance quasi-statique peut être modélisé pour des densités moins importantes. En pratique, on peut supposer qu'à pression constante, le gradient radial de densité électronique sera identique à celui de la Figure 4 et qu'une variation de puissance de la source ICP permettra de réduire n e (0) et donc de retrouver une condition de résonance sur la permittivité relative du plasma à ces fréquences.

Sur la Figure 6, trois cas additionnels à plus faible densité (d'un ordre de grandeur) ont été traités et mettent en évidence la résonance quasi-statique. On constate en effet une forte amplification du coefficient dipolaire B OU T pour des fréquences spécifiques. Lorsque n e (0) augmente, la fréquence de résonance augmente également. Ce comportement permet d'envisager l'agilité en fréquence du résonateur à plasma. Il est important de noter que les densités électroniques mesurées ici proviennent d'une mesure de Langmuir selon r dans la direction horizontale que l'on a ensuite extrapolée selon θ et φ. La disposition de la source ICP sur la partie inférieure laisse envisager que le gradient radial dans la direction verticale sera beaucoup plus important, du fait de la décroissance forte du champ selon cette direction (c'est-à-dire lorsqu'on s'éloigne de la source). Ce nouveau gradient tendra donc à diminuer la densité électronique réelle dans l'enceinte.

Modélisation numérique de l'antenne miniature à plasma

En pratique, cette discrétisation devient nécessaire lorsqu'on souhaite effectuer des simulations full-wave afin d'optimiser la source microonde (un monopole dans notre cas) au système plasma résonant [START_REF]Preliminary study on the feasibility of a plasma-based electrically small eng antenna[END_REF]. La Figure 7 compare les fréquences de résonance obtenues par le calcul analytique et la simulation sur Ansys HFSS lorsque N = 16 et pour différents n e (0). Il est intéressant de remarquer que plus n e (0) augmente, plus l'estimation de f res est mauvaise car le paramètre ka n'est plus négligeable : l'hypothèse quasi-statique n'est donc plus valide. Les paramètres d'impédance et de rayonnement d'un monopole de longueur l = 90 mm et de rayon r = 8.5 mm placé au centre de l'hémisphère de plasma et relié à un port d'excitation 50 Ω ont été calculés numériquement. Lorsque le plasma est éteint, le monopole miniature est fortement capacitif et non-résonant (sa résonance propre a lieu à plus haute fréquence). A l'inverse pour un plasma dont le gradient est celui de la Figure 4 avec une densité n e (0) = 5 × 10 8 cm -3 , une résonance apparait.

Le tableau 1 résume les différents paramètres antennaire obtenus en simulation pour un plasma éteint et un plasma allumé. Le plasma permet de réduire la fréquence de résonance du monopole de 700 MHz à environ 139 MHz. A cette nouvelle fréquence, le facteur de miniaturisation est ka = 0.313, ce qui correspond à une structure résonante de dimension caractéristique λ/20.

Conclusion

Dans ce papier, nous avons présenté une nouvelle implémentation de la résonance quasi-statique ENG pour le rayonnement d'antenne à l'aide d'une décharge plasma. Un prototype expérimental a été conçu et construit afin de valider ce concept.

Des mesures de densités électroniques ont permis d'obtenir les paramètres électriques du plasma afin d'étudier analytiquement et numériquement l'influence d'un plasma réaliste sur une structure hémisphérique. La prévision de la fréquence de résonance de la structure par le modèle analytique est cohérente avec les simulations électromagnétiques classiques effectuée sur le logiciel commercial Ansys HFSS.

Les paramètres antennaires simulés mettent en évidence la résonance quasi-statique lorsque le plasma est allumé et permettent une augmentation relative de la puissance rayonnée d'environ 18.7 dB. En outre, la structure devient furtive lorsque le plasma est éteint et le contrôle du plasma (à travers sa densité) apporte l'agilité en fréquence à cette solution.

Figure 2 .

 2 Figure 2. Schéma de conception du prototype antennaire à résonance ENG.

Figure 3 .

 3 Figure 3. Prototype expérimental avec la sonde de Langmuir montée sur le translateur linéaire.

Figure 4 .

 4 Figure 4. Profil radial de densité électronique mesuré, lissé et interpolé par un polynôme de degré 3 en r.

Figure 5 .

 5 Figure 5. Schéma du problème de la diffusion d'une sphère de plasma non-uniforme entourée de verre plongée dans un champ E 0 uniforme.

  = 0.5 × 10 9 ne(0) = 1 × 10 9 ne(0) = 1.5 × 10 9

Figure 6 .

 6 Figure 6. Coefficients dipolaires B OU T normalisés par λ 2 en fonction de la fréquence.

Figure 7 .

 7 Figure 7. Comparaison des fréquences de résonance obtenue par le calcul analytique et par la simulation électromagnétique.

  Tableau 1. Coefficient de réflexion et efficacité totale simulés avec et sans plasma

		S 11 (dB)	η tot (%)
	Plasma OFF	-0.0034	0.097
	Plasma ON	-10.11	7.1693

XX èmes Journées Nationales Microondes, 16-19 Mai 2017 -Saint-Malo

Le calcul de la fréquence de résonance f res par le modèle analytique continu sert également de référence dans la recherche du nombre minimum N de couches concentriques à considérer dans une représentation discrète du milieu plasma.

Remerciements

Les auteurs souhaitent remercier la DGA pour son soutien financier.


Keywords: Bayesian inference, Order effects, Decision-Making, Emotions, Emotional Intelligence

Bayes' inference cannot reliably account for uncertainty in mental processes. The reason is that Bayes' inference is based on the assumption that the order in which the relevant features are evaluated is indifferent, which is not the case in most of mental processes. Instead of Bayes' rule, a more general, probabilistic rule of inference capable of accounting for these order effects is established. This new rule of inference can be used to improve the current Bayesian models of cognition. Moreover, it should play an essential role in the search for artificial emotional intelligence.

Introduction

It has already been shown that the paradigm of Bayesian rationality in cognitive science is faced with some general challenges, like the problem of connecting its probabilistic models to psychological mechanisms or that of justifying the priors used in Bayesian inference [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF] (Griffiths et al. 2008). However, beyond these challenges, Bayesian rationalism suffers from an even more fundamental problem: Bayes' inference cannot account for mental processes because the latter give rise to order effects. Order effects in mental processes denote the fact that the order in which the relevant mental features are evaluated is not indifferent. Being understood as contextual or as subjective effects, these order effects cannot be accounted by the classical, set theoretical, Kolmogorov probability calculus from which Bayes' rule is derived. These non-classical effects are actually quite similar to those encountered in the quantum realm when measuring non-compatible observables, which means that they can be accounted for within the mathematical framework of quantum theory. In quantum theory, the measurable properties of any system, its "observables", are represented by Hermitian operators of a non-commutative C* algebra acting on the vector space of the possible states of this system. The only probability measure that can be assigned to their possible values is a measure over the subspaces of this vector space [START_REF] Gleason | Measures on the closed subspaces of a hilbert space[END_REF] and not as a measure over subsets of the set of possible states -like in classical probability theory. On the basis of the quantum probability calculus so defined, a new paradigm of cognition, called "quantum cognition", has been developed for a few decades by several authors, including Aerts, Sozzo, Busemeyer, Bruza, Wang, Atmanspacher, Römer and Pothos [START_REF] Aerts | Quantum structure in cognition: Fundamentals and applications[END_REF] [START_REF] Aerts | Quantum Structure in Economics: The Ellsberg Paradox[END_REF] [START_REF] Busemeyer | Quantum models of cognition and decision[END_REF]) [START_REF] Wang | A Quantum Question Order Model Supported by Empirical Tests of an A Priori and Precise Prediction[END_REF] [START_REF] Atmanspacher | Order effects in sequential measurements of noncommuting psychological observables[END_REF] (Pothos and Busemeyer 2019) [START_REF] Busemeyer | Is there a problem with quantum models of psychological measurements[END_REF]. Quantum cognition can deal with the non-classical effects that are inherent to cognition and decision making, namely, order effects and interference effects, and then with all manifestations of these effects, like the "and fallacy", the "or fallacy" or Ellsberg paradox that regards decision making in uncertain situation [START_REF] Busemeyer | Quantum models of cognition and decision[END_REF]. For our purpose, the essential point is that the study of cognitive processes requires accounting for the non-commutativity of mental observables, which thus requires working within the paradigm of quantum cognition where Bayes' rule is no more valid.

However, insofar as Bayes' rule plays an essential role in the current models of cognition and decision making, it would be rather problematic to drop this mathematical tool definitely instead of trying to generalize it in order to make it also applicable to non-commutative observables. This article proposes such a solution: the general idea of Bayesian rationality is kept, namely the idea of modeling cognitive processes in probabilistic terms, by updating the prior distribution when a new event occurs, while Bayes' classical rule is replaced by a more general probabilistic rule capable of dealing with these non-classical effects. This change is absolutely required as soon as subjective experience is involved in the mental processes considered, that is, quite in all of them as soon as their reduction to pure mechanical reasoning is inappropriate. This change is all the more necessary insofar as Bayes' rule plays a fundamental role in the current research in artificial intelligence, for categorization tasks, and in the simulation of emotional intelligence.

We will first recall, in section 1, what is Bayesian rationality and emphasize the fact that Bayes' rule on which it presently relies does not hold in the case the considered mental processes give rise to order effects. Two paradigmatic successful applications of Bayes' inference will also be presented in order to analyze the reason of this success. In section 2, we will present the order effects inherent to decision making and to all mental processes involving emotional experience. In section 3 we will evaluate some currently proposed Bayesian models of cognition: are these models really successful, as too easily claimed by the supporters of the "all-Bayesian" rationality? We will explain that this apparent success has to be questioned when the commutativity of the considered observables is not satisfied. Section 4 will present the theoretical framework of quantum cognition, an alternative approach to cognition and decision making capable of accounting for these non-classical effects, which are indeed inherent to all mental processes. Within this generalized framework, we will propose an alternative to Bayes' rule where the conditional probabilities P(Ai/Ej) are not linearly related to the "inverse" conditional probabilities P(Ej/Ai), like in Bayes' rule. This new probabilistic rule of inference, whose classical limit, when there is no order effects, is Bayes' rule, requires to compute the commutator of the couples of observables respectively defined from the series of events {Ai} and {Ej} in order to take into account their degree of non-commutativity. Section 5 will illustrate the fertility of this alternative, probabilistic rule of inference for modelling cognitive processes. It will discuss the way of improving the current models of visual perception and those of emotion recognition from the subject's face expression. Section 6 will explain why such a generalized, probabilistic rule of inference should play an essential role in the field of artificial intelligence, in particular for simulating the emotional aspect of intelligence.

Bayesian rationality and its limitations

Bayesian rationality

Bayesian rationality is an approach to cognition based on the idea that cognitive processes, and in particular reasoning, can be rationally modeled by appealing to probabilities, used as a tool to make predictions, rather than by logical rules [START_REF] Griffiths | Bayesian models of cognition[END_REF] [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF]. Many works have been developed in order to model high-level cognitive processes in probabilistic terms, in the range of perception [START_REF] Knill | The Bayesian brain: the role of uncertainty in neural coding and computation[END_REF] [START_REF] Mamassian | Bayesian modelling of visual perception[END_REF], categorization [START_REF] Ashby | Categorization as probability density estimation[END_REF], (Griffiths et al. 2008) [START_REF] Perfors | Learning to learn categories[END_REF] or language processing [START_REF] Xu | Word Learning as Bayesian Inference: Evidence from Preschoolers[END_REF], among others. It is moreover assumed that the good probabilistic calculus to use is the classical one, from which Bayes' rule is derived. Bayes' rule extends the too narrow framework of binary classical logic by taking into account the uncertainty in the knowledge of premises and the acquisition of information, which is evaluated in terms of probabilities. Even when probabilities are interpreted subjectively, as degrees of belief (de [START_REF] Finetti | Logical foundations and measurement of subjective probability[END_REF], Bayesian reasoning still satisfies the rules of the classical probability calculus, as shown by Cox-Jaynes theorem [START_REF] Cox | Probability, Frequency, and Reasonable Expectation[END_REF], which strengthens the idea that Bayesian inference is appropriate for modeling mental (or even brain) processes.

In the literature about cognition, Bayes' rule is thus regarded as the key ingredient used for integrating uncertainty in cognition and decision-making (Griffiths et al. 2008) [START_REF] Oaksford | Bayesian rationality: The probabilistic approach to human reasoning[END_REF] [START_REF] Evans | Uncertain deduction and conditional reasoning[END_REF] [START_REF] Cruz | Bayesian reasoning with ifs and ands and ors[END_REF] [START_REF] Mamassian | Bayesian modelling of visual perception[END_REF]. It is used to model many areas of human activities, like finance (for modelling risk), medicine (for diagnostic and decision making) or meteorology (for weather forecasting), and the success of Bayesian models of cognition seems certain, as well in reasoning, learning or making decision. Accordingly, Bayesian networks, based on Bayes' rule, are used in machine learning whose applications have been developed in image processing [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF], neuroscience 1 (Poggio 2016) [START_REF] Mamassian | Bayesian modelling of visual perception[END_REF] and medical diagnostics [START_REF] Kubota | Artificial intelligence used to identify skin cancer[END_REF], among other domains. Bayes' inference is supposed to correctly represent the way we reason, we learn and make decision in uncertain situation, and therefore to be a key-ingredient for developing artificial intelligence. Let us now examine accurately the presumed success of Bayes' inference and show that its application to mental processes suffers from important bias.

Bayes' inference and its domain of validity

Bayes' inference allows to calculate how a priori probabilities are updated when new information is gathered (Hossein 2014): the posteriori probability P (A/E) of the event A given the evidence E, which can denote in particular the observation of some feature or some 1 Reducing mental processes to brain's activity leads to the idea that Bayesian inference "takes part of the automatic and unconscious, elementary operations of our brain" [START_REF] Dehaene | course on the Bayesian Brain[END_REF]. According to a growing trend in theoretical neuroscience, the human perceptual system could thus be modeled as a Bayesian machine. The brain is supposed to represent sensory information probabilistically, in the form of probability distributions. This hypothesis, which presently lacks of experimental confirmation [START_REF] Knill | The Bayesian brain: the role of uncertainty in neural coding and computation[END_REF], is only an overinterpretation of Bayesian rationalism within a reductionist materialism. It will not be examined further in this article, which focuses on the rationality of mental processes.

property of the situation under consideration, is calculated from the prior probability P (A), estimated before the occurrence (or the knowledge of the occurrence) of E, and the "likelihood" P (E/A), which is the conditional probability of observing E when A is realized:

(1) P (A/E) = [P (A). P (E/A)] / P (E),

where P(E), which appears in the denominator of this expression, is the probability of occurrence of the event E, which can for example denote the observation of a property and is a priori computed independently of A.

Bayes' rule is derived from the definition of conditional probability in the classical probability calculus: P (A/E) = P (A and E) / P (E),

where "A and E" has no temporal connotation, meaning that A and E can occur in any temporal order or be simultaneous. This order-independent definition gives rise to the "rule of multiplication" of the classical probability calculus:

(2) P (A and E) = P (A/E) . P (E) = P (E/A). P (A), and Bayes'rule (1) is then straightforwardly obtained by dividing the two terms of the second equality of (2) by P(E).

However, it is essential to notice that (2) is valid on the condition that the value of the joint probability P (A and E) is independent from the order of occurrence of A and E. If P (A and E) depends on the order of occurrence of A and E, we must clearly distinguish the calculation of P (A and then E) = P (E/A). P (A), from that of P (E and then A) = P (A/E). P (E), and in this case Bayes' rule (1) is not verified since (2) is no more valid. This means that Bayes' inference is valid only if the condition of commutativity of events is satisfied.

Two paradigmatic applications of Bayes' inference

In order to check the latter condition of commutativity in the use of Baye's probabilistic inference, let us mention two paradigmatic applications of it. The first one, presented in any text-book as an illustration of Bayes' rule, describes the drawing of balls in two urns. The second application of Bayes' rule, which is widely used in the medical field, regards diagnostic testing.

Drawing of balls in two urns

We have two urns I and II that respectively contain 2 yellow balls and 6 blue balls (for urn I) and 3 yellow balls and 9 blue balls (for urn II). A blue ball is drawn from one of these two urns but we do not know which one it is2 . The problem is to evaluate the possibility that this ball was drawn from urn I or from urn II. Bayes' rule allows us to calculate these probabilities. The probability that this ball was drawn from urn I is: P (I/Blue) = P (I).[P(Blue/I) / P(Blue)],

and, interpreting here probabilities as proportions (for example, P(I) = proportion of balls in urn I = 8/20 = 0.4), leads to the result P(I/Blue) = 0.4.

This result can be checked by merely counting the proportion of blue balls in urn I, which are in number of 6 for a total of 15 blue balls in urns I and II. This proportion is exactly 0,4 -which confirms the previous result. A quite similar reasoning can be done for evaluating P(II/Blue). No dispute can be made to this reasoning based on Bayes' rule (1) since the variables that are measured, called "observables" in the following of this article, are purely objective data, contextually independent and completely independent from any subjective interpretation and any judgment. These observables are intrinsically defined as proportions of yellow and blue balls in urn I and II, that is, as objective properties of the physical world. As a consequence, the order in which they are evaluated does not matter and Bayes' rule can be successfully applied.

Contamination screening

The second example of successful application of Bayes' rule is the classic case of diagnostic testing [START_REF] Broemeling | Bayesian Methods for Medical Test Accuracy[END_REF]) [START_REF] Klement | The Epistemology of a Positive SARS-CoV-2 Test[END_REF]. We want to know if someone is infected by a virus or not by performing a test whose response is positive or negative. The events "to be infected" and "not to be infected" are respectively noted as (V+)

and (V-). The positive and negative results of the test are respectively noted as (T+) and (T-).

Suppose that we know the sensitivity of the test P(T+/V+), which is the probability that the test is positive for an infected person, its specificity P(T-/V-), which measures the reliability of the negative tests, and the contamination prevalence P(V+), which estimates the proportion of persons infected in the population. If a person has obtained a positive test, what is the probability P(V+/T+) that this person is contaminated?

Bayes' rule allows us to calculate this probability: P (V+/T+) = P(V+).[ P(T+/V+) / P(T+)],

where P(T+) can be computed as: P(T+) = P(T+/V+).P(V+) + P(T+/V-).P(V-), with P(V-) = 1 -P(V+) and P(T+/V-) = 1 -P(T-/V-).

For example, if the reliability of the diagnostic test is 0.8 for the sensibility P(T+/V+) and 0,7 for the specificity P(T-/V-), while the prior probability of contaminated people (defined as a frequency) is known to be 0.01, Bayes' rule computes that P(V+/T+) = 0,026.

This result (which shows that the test is not very significant) can be considered as valid because:

(i) the two observables that respectively measure the existence or the absence of the contamination (V = V+ or V-) and the positivity or the negativity of the test (T = T+ or T-)

are supposed to measure purely objective data or states of the world, like in the previous example (for the number of yellow and blue balls) and that, consequently, (ii) the act of observing the state of contamination of a subject (for example, by referring to clinical signs, medical imaging or biological anomaly) has no influence on the result of the test performed on her, and, reciprocally, it is granted that performing this test and stating its result does not change the state of contamination of the patient, which is regarded as an intrinsic attribute. Hence the commutativity of the observables that respectively measure the state of health of a patient (responding to the question: is this patient infected or not by the virus?) and its contamination rate (is this patient positive or negative to the test?).

However, as will be emphasized in the next section it is not the case for cognitive processes: the condition of commutativity of cognitive observables is generally not fulfilledwhich seriously questions the reliability of the current Bayesian models of cognition.

Order effects

As explained above, the commutativity of observables that are involved in the previous examples (section 1.3) relies on the assumption that the measured properties are intrinsically possessed by the entities considered (objects, animals, patients, …). However, as will be explained below, it is generally not the case of mental observables whose values are highly contextual and subject-dependent. Let us investigate this essential characteristic of mental activity on two paradigmatic examples.

Non-commutativity in decision-making

In a survey realized in 1997 (September 6-7) and involving 1002 respondents, half of the participants were asked the two questions 'is Clinton honest and trustworthy?', noted as A hereafter, and then 'is Gore honest and trustworthy?', noted as B hereafter, while the other half were asked the same pair of questions in the opposite order. As reported by [START_REF] Moore | Measuring new types of question order effects[END_REF], the list of answers for the two groups shows that Clinton received 50% agreement when asked first (which defines the "non-comparative" context) but 57% when asked second (which defines the "comparative" context because this answer can be influenced by the first one). It also shows that Gore received 68% when asked first and 60% when asked second.

This difference in the frequencies of the respondents' answers shows that the order in which the questions are asked is significant since the frequency of the positive answers to the same question depends on whether this question is asked first or second. Focusing for example on positive answers for both questions A and B, respectively noted as Ay and By, this question order effect can be expressed by the following difference: P (Ay By) ≠ P (By Ay),

where P (Ay By) is the probability of responding "yes" to question A followed by "yes" to question B, and P (By Ay) is the probability of obtaining the same answer to these questions asked in the inverse order. Moore calls this type of question order effect "consistency effect"

to denote the fact that the difference between the probabilities of positive answers for questions A and B decreases from the non-comparative context to the comparative context, which is here the case since in the non-comparative context p(By)p(Ay) = 18% while in the comparative context p(Ay/n By)p(By/n Ay) = 3%. Note that other types of order effects in decision making have been observed in other similar survey experiments, for example a "contrast" order effect showing that, unlike the previous consistency effect, the difference between these probabilities is amplified in the comparative context.

Non-commutativity of emotional observables

Order effects do not only occur in decision making but in all mental activity where subjective experience is involved. In particular, significant order effects can be observed and quantified in the domain of emotions, which should then be taken into account in the cuttingedge research in artificial intelligence (see section 6). As can be observed in daily life, felt emotions cannot be regarded as intrinsic features of a person since they are continuously changing according to our life experience. Their nature and their intensity is highly contextual since they strongly depends on our personal past and present experience of life, on our social environment and even on what we felt just a moment before [START_REF] Stolorow | The Contextuality of Emotional Experience[END_REF]. For example, asking a subject about her degree of happiness and asking the same question after reminding her of a sad event in her life generally provides different results. As was the case for the previous example of surveys with two successive questions, the order effects relative to emotions can be evaluated from data on successive measurements of the intensities of emotions felt by subjects. These intensities can be collected by asking them to report discrete values on a graduate scale (Bachorowski and Braaten 1994) or to report them continuously, using a continuous response digital interface on which the subject moves a stylus or finger [START_REF] Geringer | A fifteen-year history of the continuous response digital interface: Issues relating to validity and reliability[END_REF].

For illustration, consider the following table, drawn from an article by Prkachin and team (1999). This table reports the average intensity of five emotions experienced by subjects conditioned in target emotional states 3 by Lang's method [START_REF] Lang | A bio-informational theory of emotional imagery[END_REF]).

(from [START_REF] Prkachin | Cardiovascular changes during induced emotion: an application of Lang's theory of emotional imagery[END_REF] Let us for example focus on the couple of emotions Anger and Fear. From this table the probabilities of the sequence of evaluation "Fear and then Anger" and of the reverse sequence can be evaluated as follows: 1) from the first line (neutral target), the prior probabilities P(A) and P(F) of respectively experiencing Anger and Fear, can be computed as the ratio of the average intensity of each of these emotions (respectively 0.03 and 0.09) to the sum of all the five average intensities (0.54); 2) from the third line (Anger conditioning), the conditional probability P(F/WA) of experiencing Fear for a subject conditioned in Anger state WA can be computed. In first approximation, which can here be done in order to show the existence of order effects, WA will be identified to the target emotional state 4 A. The conditional probability P(F/WA) ≈ P (F/A) can then be evaluated as the ratio of the average intensity of Fear (0.39) to the sum of the five average intensities on this line (7.56); 3) similarly, from the fourth line, the conditional probability P(A/WF) of experiencing Anger for a subject conditioned in Fear state can be evaluated as the ratio of the average intensity of Anger (0.73) on the sum of the five numbers of this line (7). The resulting probabilities are P (A F) = P(A) x P(F/A) = (0.03/0.54) x (0.39/7.56) = 2.87 10 -3 P (F A) = P(F) x P(A/F) = (0.09/0.54) x (0.73/7) = 1.74 10 -2

This clearly shows a net difference (of a factor of about 6) between the sequential probabilities P (A F) and P (F A).

It can also be shown that emotional observables do not commute neither with their physiological correlates nor with their behavioral correlates. This point can be established from data about the joint measurement of emotional observables and their physiological or behavioral correlates reported in the literature (Barrett et videos showing the participants performing the required tasks, by noting for example their facial expression and their body movements. Kassam's and Mendes' study clearly shows that for these subjects, conditioned in angry state, the changes in the values of the cardiovascular and behavioral observables are significantly different depending on whether or not they report their emotional state (of anger, in this example). This tells us that the successive measurement of emotional and physiological or behavioral observables gives rise to order effects since if these observables were commuting their values would not be inter-dependent, as is actually the case.

A more precise characterization of these order effects in emotional life requires a mathematical representation of the non-commutativity of the emotional observables with each other and with their physiological and behavioral correlates. This representation will be proposed in section 5 and Appendices 1 and 2.

Are current Bayesian models of cognition reliable?

As shown in section 2, mental processes can give rise to order effects. However, as emphasized in section 1, Bayes' rule is valid only under the assumption of commutativity of the observables involved in this probabilistic inference, which means that order effects cannot be taken into account. So, we can be suspicious about the validity of the current Bayesian models of cognition. Are these models reliable? Let us examine some of them.

Bayesian models of categorization

The ability to classify objects (or concepts) into categories is a fundamental cognitive task realized by humans in order to organize their mental representation of the world they live in and thus to cope with it. "Categories" form the basic cognitive mental representations in which and by which humans can organize their knowledge and make inferences about the world [START_REF] Murphy | The role of theories in conceptual coherence[END_REF]) [START_REF] Rosch | Principles of categorization[END_REF]. They group concepts which share similar properties, like their form, their color or their behavior. Categorization is the act of classifying a new object according to these categories. A Bayesian model of categorization computes the probability of classifying it into each of the categories, given the properties of this object, and determines the most probable category label it can be assigned to.

Category learning models aim to explain how category labels can be learned, for example by children. They are based on the same Bayesian method of classification of objects or concepts into appropriate categories and aim to explain that the acquired knowledge is used to make decisions about how to categorize new stimuli. Several rational analyses of category learning have been proposed [START_REF] Perfors | A tutorial introduction to Bayesian models of cognitive development[END_REF]) (Ashby and Alfonso-Reese, 1995) [START_REF] Nosofsky | Attention, similarity, and the identification-categorization relationship[END_REF]. All these models compute via Bayes' rule the probability distributions associated with different category labels and select the most probable result.

However, like for any Bayesian model, these categorization models require defining the set of "primitive" properties for which the prior probability distribution and the likehood function are provided. This is where the crucial question arises: can these properties be intrinsically defined, and therefore commute with each other, or not? In the current Bayesian models of categorization or category learning this condition is satisfied only if these "primitive" properties can be regarded as purely objective ones on which anyone agrees, being of physical, biological or behavioral order. This is actually the case in most of the current Bayesian models of categorization. For example, in the model of category learning presented by [START_REF] Perfors | A tutorial introduction to Bayesian models of cognitive development[END_REF], classifying a new item in the categories "dog", "fish", "bird", "tree", flower", fruit" involves the observation of some of its physical properties on which anyone agrees, like its size, its color, its hairiness or skin appearance (a crucial property to classify animals), or the fact it has leaves (a property of trees).

However, this kind of categorization based on intrinsically defined (and then commuting) properties covers only a small part of the real human's experience of categorization. As can be observed in everyday life, and as emphasized in section 2, mental activity generally involves contextual and subject-dependent observables, which do not commute with each other. Think about our very selective ability to perceive the world around us, this selective perception of the world being a form of subjective bias in the interpretation of information in order to fit with our values and our beliefs [START_REF] Pronin | Perception and misperception of bias in human judgment[END_REF]. This subjective tonality of cognitive processes has been observed for a long time by psychologists for "normal" as well as for pathological subjects [START_REF] Cattell | The subjective character of cognition and the pre-sensational development of perception[END_REF]) [START_REF] Roiser | Hot and cold cognition in depression[END_REF].

Psychoanalysts go even so far as to talk about "resistance" to accessing memories, which is a form of cognition, when the latter are associated to traumatic feelings [START_REF] Freud | Cinq leçons sur la psychanalyse, deuxième leçon : Résistance et refoulement, in Leçons d'introduction à la psychanalyse[END_REF]. It seems then difficult to ignore the role of emotions in cognitive processes. As a consequence, the observables involved in categorization tasks do not generally commute.

An important example of such non-classicality is provided by the task of assigning emotional states (which play the role of categories) to human subjects from the observation of their facial expression. Computing the probability that a subject be labeled as "sad", "happy"

or "angry" given the observation of some features of her face expression must take into account the non-commutativity of the relevant (emotional and behavioral) observables, which gives rise to some unavoidable uncertainty in the correspondence between a subject's facial expression and the nature (and/or the intensity) of the experienced emotion -uncertainty which has been noticed by several authors (Barrett et al. 2017) (Duran and Fernandez-Dols 2019). This paradigmatic example will be developed in section 4 with the appropriate mathematical tools.

Bayesian models of visual perception

The quasi-impossibility to separate subjective experience from the idealized "cognition" of intrinsic properties of the world has also been observed in visual perception, even if this subjective component relies on less personal features. In the current Bayesian models of visual perception [START_REF] Knill | The Bayesian brain: the role of uncertainty in neural coding and computation[END_REF]) [START_REF] Kersten | Bayesian models of object perception[END_REF] ( [START_REF] Mamassian | Bayesian modelling of visual perception[END_REF], the task to accomplish is to predict the most probable perceived image given the prior probability distribution of physical properties, like reflectances, shape and orientation of surfaces, object size or wavelength, and the likelihood function that a given set of values of these physical properties generate a particular image. The "primary"

properties are then defined as intrinsic features of the physical world, as can be checked for example in the 3-D model of visual perception proposed by Mamassian and team (2002). A few solutions for tackling this problem has been proposed. [START_REF] Mamassian | Bayesian modelling of visual perception[END_REF] suggest to allow the components of Bayes' inference, in particular the likelihood function and the prior distribution, to vary from trial to trial, that is, to be stimulus-dependent, in contrast with the strict Bayesian method where the latter are supposed to be intrinsically defined (see section 1). [START_REF] How | Comparison of Bayesian and empirical ranking approaches to visual perception[END_REF] have suggested to explicitly take into account the context of perception, which can be illustrated by the mention of the white surrounding or the black surrounding in the previous example. Taking into account the context could be implemented by trying to compute the posterior conditional probability distribution for the target and the surround, which, as emphasized by How [START_REF] How | Comparison of Bayesian and empirical ranking approaches to visual perception[END_REF], section 7), seems rather complicated to implement for complex contexts. These authors have proposed to take into account the contextual effect within their "empirical ranking approach", by computing the probability distribution associated to the values of the considered properties (the target luminance, for example) that have co-occurred with each of the possible contexts.

However, these interesting attempts to improve Bayesian models of visual perception ultimately result in models that are not any more strictly "Bayesian" since the properties on which the Bayesian inference bears cannot be defined intrinsically, as properties of the physical, objective world -like in the paradigmatic uses of Bayes' rule presented in section 1.3. The resulting models actually deal with non-commutative observables, this noncommutativity being nothing but the very expression of the fact that the relevant properties to consider in visual perception (being understood as contextual, stimulus-dependent or subjectdependent) cannot be intrinsically defined.

As shown by the previous developments, the Bayesian models of cognition, which rely on categorization tasks, must be revisited in order to account for the order effects inherent to any mental process. This change can be done elegantly by deriving an appropriate probabilistic rule that generalizes Bayes' rule. This new rule will be derived in the next section within the mathematical framework of quantum cognition, which will be first presented.

Integrating order effects in Bayesian models of cognition

Theoretical framework

Integrating order effects in Bayesian models of cognition requires leaving the classical probability calculus since the latter cannot account for them. It requires appealing to a generalized probability calculus where the probability of a sequence of events P(A and then B) can be distinguished from that of the reverse sequence P(B and then A). Such a generalized probability calculus already exists, it has been built by physicists in order to deal with quantum phenomena, which generally give rise to order effects, and it has already been applied in the field of cognition and decision-making. "Quantum cognition" thus deals with cognitive processes within the same mathematical framework than that of quantum theory. It │> is the state of the subject, PAi the projector on the eigenspace associated to Ai, which, for sake of simplicity, has been assumed to be one-dimensional, spanned by the vector │Ai> of H. This projection can be illustrated as follows:

|| PAi|ψ〉|| By Born rule, which defines the only probability measure in such a state space according to Gleason's theorem [START_REF] Gleason | Measures on the closed subspaces of a hilbert space[END_REF], the probability P(Ai) of occurrence of Ai can be computed from the projection PAi|ψ〉 of │>, as P(Ai) = || PAi|ψ〉|| 2  <ψ|PAi|ψ>.

In this geometrical representation, the order effects relative to the successive measurement of the observables A and B with respective results Ai and Bj are thus captured by the non-commutativity of the projectors PAi and PBj :

PAi PBj ≠ PBj PAi.

Many "fallacies" in cognition and decision-making can be explained within this quantum-like approach (Busemeyer and Bruza 2012) [START_REF] Aerts | Quantum structure in cognition: Fundamentals and applications[END_REF]. For example, the "conjunction fallacy" is the fact that, in contrast with the classical probability calculus, the probability of occurrence of the conjunction of two events reported by human subjects is often greater than the probability of the occurrence of each of them. This phenomenon finds an explanation within this quantum-like framework. The probability of a sequence of events "A and then B", noted as P (A B), can be computed as || PB PA |ψ〉|| 2 , where PB and PA are, respectively, the projector on the subspace of the Hilbert space spanned by the eigenstates of This generalized probabilistic framework has also been applied to deal with decision making in uncertain situation. For example, the famous Elsberg's paradox [START_REF] Ellsberg | Risk, Ambiguity and the Savage Axioms[END_REF] bearing on the behavior of economic agents in a situation of uncertain knowledge can also be solved within this mathematical framework. Elsberg's paradox puts into question the law of total probabilities of the classical probability calculus. According to the quantum-like approach to cognition and decision making presented here, the paradoxal difference between

A new probabilistic rule of inference

Within this quantum-like framework, Bayes' rule can be generalized in order to take into account the non-commutativity of mental observables which, as seen above, is involved in most of the mental processes. As explained in Introduction, the derivation of this new rule is justified by the will to continue working in the paradigm of Bayesian rationality, while making it capable to deal with order effects. This generalization of Bayes' rule will thus greatly improve the current Bayesian models of cognition where the role of subjective experience cannot be ignored and thus paves the way to a new, more realistic, implementation of emotional intelligence. 6 The precise calculation of the relevant commutators and probabilities for this survey will not be presented here since we essentially focus on the order effects involved in cognitive processes (and not in decision making). 7 Also note that another characterization of this order effect has been proposed more recently by [START_REF] Busemeyer | Is there a problem with quantum models of psychological measurements[END_REF], by referring to a so-called "ABA experiment", where a measurement of B is inserted between two measurements of A. It shows that the second measurement of A can be different from the initial one and the degree of incompatibility of A and B can thus be evaluated from this difference. This order effect has been checked on the sequence ABA with a population of 325 participants on a wide range of 12 different set of issues.

projectors associated to the event E and F are commuting, and then when <Q >ψ = 0, is nothing but Bayes' rule (1): P (F/E) = [P (E/F) x P (F)] / P (E).

Bayesian models of cognition revisited

As noted above, the current Bayesian models of cognitive processes are based on categorization tasks, which are modelled on the basis of Bayes' inference. In order to take into account the order effects inherent to most of mental processes, these models should then be revisited. Still keeping the Bayesian approach to cognition (see section 1.1), this change only regards the inference rule that must be used to update the probabilities of realization of some assumption when new information is gathered. These models must use the general probabilistic rule (5) instead of Bayes' rule (1) insofar as the latter cannot account for order effects. Such a change thus requires to first compute the commutators of the couples of observables involved in these models.

The Bayesian models of perception involve two types of observables that do not commute: those characterizing the subject's perception and those characterizing the physical situation. A typical example of observable of the first type is the observable "Luminosity", noted as L in the following, which measures the perceived luminosity of an object by the subject. The observables of the second type measure the relevant physical features of the object or those of the context in which this object is perceived, noted as  i . The commutator of the couples of observables (L,  i ) can be computed from data relating the change of the perceived luminosity of an object to the change of its physical features, which allows us to compute the likehood function. However, such data, which could be easily gathered by asking subjects to report on a scale the perceived luminosity corresponding to different values of the physical observables  i seem to be missing for now. Consequently, we will not deepen here these calculations for models of perception but this method will be applied below to other categorization models for which more data are available. Moreover, we would like to focus on the Bayesian models of cognition that explicitly involve experienced emotions for their central role in the search of emotional intelligence. The widely spread (mis)use of algorithms supposed to detect the experienced emotions of subjects from their behavior, in particular from their facial expression, deserves some priority.

Assigning an emotional state to a subject requires finding the greatest conditional probability that the subject could be classified in this emotional state given the knowledge of some of her behavioral or physiological features. In more technical language, this task requires computing the conditional probabilities P (E k / { i } and {Bj}) that the emotional observable takes the value E k given the knowledge of the values of a set of physiological observables { i } and those of a set of behavioral observables {Bj}. However, as shown above (sections 2.2), emotional observables do not commute with each other and do not commute neither with their physiological nor with their behavioral observables. Classifying emotions from the knowledge of behavioral and/or physiological features thus requires to take into account these order effects, which means that the generalized probabilistic rule (5) must be used instead of Bayes' classical inference rule. This can be done by first computing the relevant commutators. Within the quantum-like model presented in section 4, an emotional observable, which is formally defined as a projector acting on the vector-space of all possible mental states, measures the intensity of a specified emotion experienced by the subject. As mentioned in section 2.2, this measurement can be realized by asking the subject to report discrete values on a rating scale or to report them continuously, for example by using a continuous response digital interface.

The commutators of couples of emotional observables measure their degree of incompatibility. A method of computation of these commutators from the data collected by Pkachin et al. (1999) has been proposed by Uzan in reference [START_REF] Uzan | Complementarity in Psychophysics[END_REF]). In the case analysed in [START_REF] Uzan | Complementarity in Psychophysics[END_REF] the experimenter tells the subjects stories that are directly related to these specific incidents, but other methods based on the projection of films have also been shown reliable [START_REF] Kassam | The effects of measuring emotion: physiological reactions to emotional situations depend on whether someone is asking[END_REF]. The five considered emotional contents are here "happiness" (noted as H), "sadness" (S), "anger" (A), "fear" (F) and "disgust" (D). The commutator of a couple of emotional observables (A, B), which are projectors, can be computed if the conditional probabilities that an individual be "observed" (through a questionnaire or other means) in the emotional state A if she/he has been conditioned or "prepared" in some specified emotional state B. The way these conditional probabilities can be computed for the table of data drawn from Prkachin's and team's article have been explained in section 2.2: the conditional probability P (A/WB) that a subject conditioned in the emotional state WB experiences the emotion A can be assimilated with the rate of the reported average intensity of A with respect to the sum of all the reported average intensities of emotions for subjects prepared in the same target emotional state B. This computation has been done from Prkachin's data that report these average intensity for each emotion.

Appendix 1 computes the commutator of the emotional observables Anger (A) and Disgust (D), which are interpreted in this model as projectors acting on the same 2-dimensional complex Hilbert space.

The degree of non-commutativity of emotional observables and their physiological correlates can be evaluated from data reporting the outcomes of joint measurements of intensity of emotions and physiological variables. This computation can be done from the data provided by Pkachin et al. (1999), [START_REF] Kassam | The effects of measuring emotion: physiological reactions to emotional situations depend on whether someone is asking[END_REF], [START_REF] Pauls | Repressive and Defensive Coping During Fear and Anger[END_REF] and [START_REF] Sinha | Cardiovascular differentiation of emotions[END_REF]. These articles report the changes in the values of several cardiovascular variables of subjects experiencing emotional states induced by stimuli designed according to reliable procedures, like for example the one proposed by [START_REF] Lang | A bio-informational theory of emotional imagery[END_REF].

In the case analysed in [START_REF] Uzan | Complementarity in Psychophysics[END_REF] the experimenter tells the subjects stories that are directly related to these specific incidents, but other methods based on the projection of films have also been shown reliable [START_REF] Kassam | The effects of measuring emotion: physiological reactions to emotional situations depend on whether someone is asking[END_REF]. From the data provided by the latter article it is possible to first compute the components of a subject's cardiovascular state (specified by her heart rate, for example, and noted as |HR>) in the 2-dimensional basis of the eigenstates |A n >, for n = 1 or 2, of an observable A which notifies whether or not the subject has reported her emotion of Anger. These components can be computed by the respective variations HR in heart rate when this subject respectively reports her emotional state (of Anger) and does not report it -these data are provided on page 3 of the previously mentioned article by [START_REF] Kassam | The effects of measuring emotion: physiological reactions to emotional situations depend on whether someone is asking[END_REF], in the section "Cardiovascular reactivity".

The commutator of the two projectors respectively associated with the measure of HR and that of Anger can then be easily computed. This commutator, which is different from the null operator, is computed in Appendix 2.

Finally, an estimation of the degree of non-commutativity between emotional and behavioral observables can be done from the very comprehensive data presented in the articles by Duran and Fernandez-Dols (2010) and by [START_REF] Barrett | Emotional expressions reconsidered: challenges to inferring emotion from human facial movements[END_REF]. These data question the common view according to which there would exist a well-established and universal correspondence between a subject's experienced emotion and her facial expression. Roughly speaking, these articles show that if subjects smile when they are happy and scowl when they are angry more often than would be expected by chance, significant variations of these correspondences exist across subjects, contexts and cultures [START_REF] Barrett | Emotional expressions reconsidered: challenges to inferring emotion from human facial movements[END_REF]. The common-view correspondences between the nature of experienced emotions and the expressive facial configurations of the Facial Action Coding System 8 (FACS) are tested by referring to meta-analysis of experimental data and they are shown to actually be rather weak.

For our purpose, which is to evaluate the degree of non-commutativity between the observables involved in these correspondences, we will focus on the conditional probabilities to detect the nature (and/or the intensity) of a subject's experienced emotion from her facial expression. Figure 8B of the article by [START_REF] Barrett | Emotional expressions reconsidered: challenges to inferring emotion from human facial movements[END_REF], which refers to the article by Duran, Reisenzein and Fernandez-Dols (2017), reports the proportion of successful correspondences between emotions (induced by presenting objects or events to the subjects) and the common view facial expression of these emotions. For example, the tested subjects assign the emotional experience of Anger to the common view facial expression of Anger (which is characterized by brows furrowed, eyes wide, lips tightened and pressed togethersee figure 2 A of [START_REF] Barrett | Emotional expressions reconsidered: challenges to inferring emotion from human facial movements[END_REF]) with a (weak) proportion of .28 with regard to the six considered main emotions of the experiment. This means that the conditional probability of detecting Anger given its common view facial expression, noted as FA, can be evaluated as P(A/FA) = .28. The commutator of the observables A and FA, which respectively evaluate the intensity of Anger and the degree of resemblance of the subject's facial expression with the common view facial expression of Anger, can then be computed within the theoretical framework presented in section 4. This computation is explicitly done in Appendix 3.

Generally speaking, the task of assigning emotions to facial expressions have to take into account the degree of non-commutativity between the relevant emotional observables and the observables that measure the subject's facial expressions.

Implementing emotional intelligence

8 The Facial Action Coding System has been developed by the psychologists Paul Ekman and Wallace Friesen in 1978. It is now the standard tool used in psycho-physiological studies of facial expression.

As is well known, human intelligence is not reduced only to the intellectual performance (or the IQ) but it has an important emotional part. Emotions play a crucial role in any aspect of life [START_REF] Wang | Culture and memory[END_REF] [START_REF] Elfenbein | On the universality and cultural specificity of emotion recognition: A meta-analysis[END_REF] and emotional intelligence is the way we deal with them. Salovey and Mayer defined emotional intelligence as [START_REF] Salovey | Emotional intelligence[END_REF]:

"…the ability to perceive and express emotions, to integrate them to facilitate thought, to understand and to reason with emotions, as well as to regulate emotions in oneself and in others".

For doing that, the human uses verbal communication, that is, speaking and writing, but also, consciously or not, non-verbal communication, like reading persons' face expression, observing body movements and postures, and also physiological manifestations.

By giving us access to our own emotions and to those of other persons, emotional intelligence allows us to develop empathy and to find the appropriate behavior in real time. Emotional intelligence thus has a fundamental social aspect since it helps us to connect our internal world to social reality by making decision based on the knowledge of the emotional tone of our social environment.

Simulating emotional intelligence algorithmically is now an important subject of research in the field of artificial intelligence, namely for improving human-machine interaction. It first requires emotion recognition, which can be assimilated to a classification task from multimodal sensory, behavioral or physiological data [START_REF] Poria | A review of affective computing: From unimodal analysis to multimodal fusion[END_REF]. Human emotions are "recognized" by using several types of sensors that detect speech signal, voice tone, facial expressions and body language, and by appealing to data on previously observed correlations (called above "the common-view correspondence") between the nature and intensities of emotions, on the one hand, and the nature of values of physiological and behavioral observables, on the other hand. Actual results based on facial expression or bodily movement recognition utilize statistical techniques, namely the use of supervised machine learning algorithms. Several deep learning algorithms compute the most probable emotional state which can be assigned to a subject by analyzing her behavior, like her facial expression [START_REF] Li | Deep facial expression recognition: A Survey[END_REF] or her body language and posture, which seems the most informative observations even realized from afar and from any angle of view [START_REF] Santhoshkumar | Deep Learning Approach for Emotion Recognition from Human Body Movements with Feedforward Deep Convolution Neural Networks[END_REF].

However, in the learning phase of these algorithms, the primitive properties that are observed on a huge number of images (like a collection of facial expressions, of postures or body gestures) are presently regarded as objective properties that could be observed independently of the possible subject's emotional state. Similarly, the likehood function, which relates them, through conditional probabilities, to the emotional categories to recognize (generally, five or seven "primary" emotions) are regarded as objective relationships, always carried out, independently of any other relevant parameter. However, as explained above (sections 2.2 and 5.2), this classical approach to cognitive processes involving emotional experience cannot be held anymore because there are significant order effects in the successive measurement of the intensity of emotions and their behavioral or physiological correlates.

Moreover, the emotion recognition task is only the first step to achieve in order to simulate emotional intelligence. Similar to the case of a physician whose diagnosis is not only based on medical imaging and blood analysis but also on thinking about the history of the patient, the best interpretation of her symptoms and their possible link, emotional intelligence requires much more capabilities than only detecting emotions. In addition to understanding our own emotions and those of our social environment, emotional intelligence requires the ability to make probabilistic inferences from these data. In particular, in order to respond appropriately to a situation, emotional intelligence requires to make predictions about the emotional state of a subject and her behavior given her present emotional state and new data, like information about the change of her face expression and her voice tone. This requires again the use of a probabilistic rule capable of accounting for the non-commutativity of emotional observables with each other and with their behavioral and physiological correlates.

Consequently, we need to use the generalized probabilistic rule (5) instead of Bayes' classical inference rule (1) in order to compute the required conditional probabilities P(Ek/{ i }), P(E i /{Bj}), or even P(E k /{ i } and {Bj}) that the presented item can be classified as being in the emotional state E k given the values, respectively indexed by i and j, of the physiological and the behavioral observables. The deep learning algorithms that realize the task of classifying face expressions in various emotional categories and predicting emotion from emotional, sensory, behavioral or physiological data, which involve measuring noncommutative observables (Singh, Majumber and Behera 2014) [START_REF] Cohen | Learning Bayesian network classifiers for facial expression recognition both labeled and unlabeled data[END_REF] can thus be improved by using the generalized probabilistic rule (5) for probability computations instead of Bayes' inference (1). This can be done the same way as in the developments presented in section 5.2, by first computing the relevant commutators.

We have to notice that the previous conclusion on the simulation of emotional intelligence indeed applies for the simulation of all aspects of mental activity, not only because this non-commutativity also holds for various other aspects of cognitive activity but also because emotions are involved in them to varying degrees. As shown by [START_REF] Wang | Culture and memory[END_REF] and as reported by well-known studies in the field of psychoanalysis [START_REF] Van Der Linden | L'émotion, ciment du souvenir[END_REF], the emotions felt during life situations connect its different aspects in our mind and thus play an essential role in their memorization. In daily life, "negative" emotions, like anger or sadness, can disturb our concentration and make it difficult remembering a telephone number or performing a simple mental calculus, while "positive" emotions, like a feeling of happiness, can improve our ability to perform these tasks. More generally, we can say that emotions play an essential role in all human activities, being of psychological, social or cultural order [START_REF] Hwang | Functions of emotions[END_REF].

Thus, it seems that the generalized probabilistic inference rule (5), which, in contrast with Bayes' rule, accounts for order effects, must be used for realistically modeling all aspects of human activity and for improving the algorithms that simulate these activities. Future research in the field of artificial intelligence and robotic should integrate this new probabilistic rule.

Conclusion

This article has questioned the "all-Bayesian" dogma defended by most researchers in cognitive science and artificial intelligence. Its central rule, namely Bayes' rule, is derived from the classical probability calculus, which, as shown in this article, cannot reliably deal with mental processes. The reason is that mental processes give rise to order effects which cannot be accounted by the classical probability calculus -and then by Bayes' rule. Keeping the general idea of Bayesianism, according to which cognitive processes can be rationally modeled by using probabilities rather than rigid rules of inference, a new, probabilistic rule generalizing Bayes' rule and capable of accounting for order effects has been proposed in section 4. The application of this new rule requires to compute the commutators of the observables involved in the studied situation, which can be done within the generalized probability theory developed by physicists to deal with quantum phenomena and briefly presented in this same section. As explained in sections 5 and 6, the use of this generalized probabilistic rule instead of the classical Bayes' rule will make the current models of cognition and the algorithms of artificial intelligence more reliable.

  However, as noted byOwe, Lotto and Purves (2006), visual perception (like any perception, indeed) cannot be exclusively based on intrinsic features of the physical world.Visual illusions show that what we see is not the "objective" reality, exclusively describable in terms of physical properties of the world. For example, in White's optical illusion the same target luminance can elicit different perceptions of color and brightness in different contexts (see figure1). This figure shows that black and white horizontal bars alternate and that shorter grey bars cover the white bars at left (A) and the black bars on right (B). Though the shorter grey bars have the same color and the same opacity in (A) and (B) and therefore reflect the same amount of light, the grey bars in (B), surrounded by white stripes appear lighter and brighter than the grey bars in (A), which is surrounded by black stripes. This means that how a subject perceives the shorter grey bars is not exclusively determined by their intrinsic, physical properties but it strongly depends on the subject taking into account the background on which they are perceived. Visual perception, like any perception, is highly contextual and subject-dependent.

Figure 1 .

 1 Figure 1. White's illusion

  has been developed for a few decades by several authors, includingAerts, Sozzo, Busemeyer, Bruza, Wang, Atmanspacher, Filk, Pothos and Wang (Aerts et al. 2011) [START_REF] Aerts | Quantum Structure in Economics: The Ellsberg Paradox[END_REF] [START_REF] Busemeyer | Quantum models of cognition and decision[END_REF] [START_REF] Wang | A Quantum Question Order Model Supported by Empirical Tests of an A Priori and Precise Prediction[END_REF] (Atmanspacher and Filk 2013),(Pothos and Busemeyer 2019) [START_REF] Busemeyer | Is there a problem with quantum models of psychological measurements[END_REF]. Its basic idea is to represent geometrically the observation of a mental feature of a subject by the action of a projector on the vector-state representing her mental state (or her belief state). The latter is defined as an element of the Hilbert vector space H of all possible mental states (or belief states) and, like in quantum theory, the observables are represented by Hermitian operators forming a non-commutative algebra and whose (real) eigenvalues are the possible results of their measurement. Within this geometrical representation, measuring the observable A and obtaining the outcome Ai is represented by the projection PAi|ψ〉, where, in Dirac notation,

Fig 1 .

 1 Fig 1. A simplified representation of the measurement of the observable A with result Ai

  Fig 2. P (B) can be smaller than P (A and B) in contrast with the classical probability calculus

  the probability of the agent's choice provided by the classical probability calculus and the experimental result comes from the interference term between the possible belief states of a participant[START_REF] Aerts | Quantum Structure in Economics: The Ellsberg Paradox[END_REF] [START_REF] Busemeyer | Quantum models of cognition and decision[END_REF] [START_REF] Uzan | Psychologie cognitive et calcul quantique. Implications Philosophiques, section Epistémologie[END_REF]. Also note that[START_REF] Wang | A Quantum Question Order Model Supported by Empirical Tests of an A Priori and Precise Prediction[END_REF] have represented the question order effects presented in section 2.1 above within this quantum-like framework. If the sequence of questions A and then B asked in the Clinton/Gore survey obtains the positive answers Ay and By, the corresponding construction in the vector space of all possible mental sates consists in the subject's mental state |ψ〉 the projector PAy onto the eigenspace associated to the answer Ay and then by applying to the result PAy |ψ〉 of this first operation the projector PBy onto the eigenspace associated with the value By. The probability of the sequence of answer Ay By is then calculated (by Born rule) as: P (Ay By) = ||PByPAy|ψ〉|| 2 = <ψ|PAyPByPAy |ψ>, while the probability of the inverse sequence (By and then Ay) is calculated as: P (By Ay) = ||PAyPBy|ψ〉|| 2 = <ψ|PByPAyPBy |ψ>. If the projectors PAy and PBy are non-commuting, which means that the commutator [PAy, PBy] = PAy PBy -PBy PAy is different from the null operator, P (Ay By) and P (By Ay) are different 6 . Order effects are then nicely and successfully represented in this geometrical construction. 7

It is supposed that the drawings are equiprobable.

Indeed, the target emotional state is not totally reached by the subject. The distinction between the latter state and her real emotional state is considered in the more precise calculations of section

and Appendix 1.

The pre-ejection period is the time elapsed between the depolarization of the left ventricle and the beginning of ventricular ejection. Its value is strongly dependent on that of the volume of blood ejected by the left ventricle at each cardiac cycle.

To compute the conditional probability P (E / F) for the mental events ot the mental properties E and F, let us first compute the difference of the sequential probabilities P(E F) -P (F E) within this quantum-like framework:

(3)

where the last equality has been obtained by factorizing the expression between the bra <ψ|and the ket|ψ>, and by using the definition of the commutator [PE, PF] = PE PF -PF PE. Defining the operator Q as:

the difference of sequential probabilities P (E F) -P (F E) can be written as the expectation value of Q in the mental state |ψ>:

(4)

P (E F) -P (F E) = <ψ| Q |ψ>  <Q >ψ.

Consequently, by the definition of the conditional probability of the occurrence of the event F

given that of the event E (or given the knowledge of E), which is here supposed to occur before F:

by the symmetrical definition of the conditional probability of the occurrence of the event E given that of F, which is here supposed to occur before E:

and by using equation ( 4) above we obtain the following new rule of probabilistic inference:

(5) P (F/E) = [P (E/F) x P (F) + <Q >ψ] / P (E).

This new generalized probabilistic rule of inference computes the conditional probability of occurrence of the event F given the occurrence (or the knowledge of the occurrence) of the event E for a subject in the mental state |ψ>. Its classical limit, when the