speak of a "South Atlantic system," a notion that has been developed by Alencastro (2018), while Thornton (1992) explores the notion of "Atlantic world." In 1983, the art historian Farris Thompson was the first to use the expression "black Atlantic" in reference to a supposed continuity between African and Afro-American cultures. Many other authors would follow him down this path, attempting to demonstrate the existence of a "common ground" linking Afro-American cultural and religious practices to their African origins. However, Gilroy's perspective is fundamentally different, as it does not attempt to highlight continuities between the "diaspora" and the "motherland," but rather the circulation of people, things and ideas within the "black Atlantic," putting in touch realities that are not necessarily the same.

 Capone, 1999a Capone, , 2004b Capone, , 2005)). If the Yoruba identity in Nigeria also needs its American "mirror" to exist, the so-called "globalization of the Orisha religion" 3 is then the product of this incessant negotiation between different versions of the "Yoruba tradition" in Africa as well as in the diaspora.

Afro-Atlantic religions and transnationalism

Yoruba-derived religions in the Americas have been more or less transnational since at least the latter half of the nineteenth century. Yoruba religious networks have been forged through travels of Afro-descendants and more recently by the hosting of international conferences, pilgrimage tourism, and online exchanges between initiates on the two sides of the Atlantic Ocean. In a previous work (Capone, 2004a), I highlighted the link between the field of Afro-American studies and the transnational approach, suggesting that, from

The Orisha religion in a transnational perspective Stefania Capone (CNRS/EHESS, Paris)

In a provocative essay, LJ Matory (2009: 240) asked what would a theory of transnationalism and globalization look like if inspired by "spirit possession and polytheism" rather than by "the ontologies and eschatologies of the Abrahamic and the karmic religions."

As an anthropologist, consecrated since the beginning of the 1980s to the study of the then-called Afro-American religions,1 I have long been interested in their transnationalization within a space of circulation including three continents: Africa, Americas and Europe. This diffusion conjures up the triangular trade's space that marked the birth of the "Atlantic World," a central notion in contemporary historiography from which other powerful metaphors emerged, such as the "black Atlantic," theorized by [START_REF] Gilroy | The Black Atlantic. Modernity and Double Consciousness[END_REF] as an intercultural and transnational formation, a space of interaction that made possible the formation of Afro-Atlantic cultures and religions. the beginning, the Afro-American universe had been structured according to a "transnational" logic and not simply by the forced displacement of enslaved Africans. In the case of Afro-Atlantic religions, the transnational approach is indeed imposed by the ethnographic materials. Today, their study must also consider the ritual networks between Africa and Brazil, or between the different centres preserving African traditions on the American soil, that have a direct impact on local contexts. We have therefore replaced the comparative approach underlining the cultural continuity between Africa and the Americas by a new one taking into account the space of circulation of social actors, religious practices, symbols and knowledge.

The points that I will raise in my Keynote Lecture are the fruit of a long personal journey going back to the beginning of the 2000s when I undertook the study of the transnational networks of Afro-Atlantic religions-particularly between Brazil, Cuba and the United States-through the analysis of the re-Africanization processes within Brazilian Candomblé and Cuban Santería or Regla de Ocha. 4 Since the 1970s, we have been challenged by the spread of the Orisha religion that obliges us to study these religious phenomena in a network, by linking different localities of the "Afro-religious" Atlantic.

Today we can no longer understand a "local" phenomenon without having a more "global" vision of its developments and its confrontations with other religious contexts, in a dialogue that is often tense between the "diaspora" and the "motherland." This led me to create in the early 2000s a research group on the transnational networks of Afro-American or Afro-Atlantic religions and, from 2008 to 2011, to coordinate a research team on the transnationalization of Afro-Atlantic religions in the Americas and Europe, within the international project RELITRANS, funded by the French National Research Agency (ANR) 5 . My reflection was thus nourished by a collective work6 that contributed to the development of research on transnationalism in France, including several doctoral theses on the transnationalization of Afro-Brazilian religions in Europe (Portugal, France, Italy, Switzerland, Austria and Germany). Today we have thus gathered enough in-depth research, carried out in different national settings, allowing us to identify some recurrent dynamics within these processes.

Transnationalism or globalization

We are all aware that, in the context of what is called "cultural globalization," religions are undergoing significant transformations. On the one hand, there is an intensification of the translocal circulation of followers, symbols, and beliefs that once belonged to a religious practice linked to a specific historical and geographical context. On the other hand, local religious fields are opening up to new practices and new representations.

Globalization has had an impact, without comparison, on the displacement of certain religious practices which, until then, had remained deeply rooted in specific traditions, territories and social groups. The use of the term "globalization" precisely reflects this level of integration and interconnection, which today is reflected in the individual's empirical perception of belonging to a "global world," beyond any real territorial attachment. Thus, being initiated into one of the Afro-Atlantic religions implies the perception of being part of a whole that goes beyond the limits of one's own daily experience in local worship communities, thanks mainly to social networks that bring together practitioners from, at least, three continents.

However, this religious globalization often implies a reversed process, from South to North, from peripheries to metropolitan centres, or, in our case, from "diaspora" to "motherland." Unlike the "globalization" of the great monotheistic religions, these shifts are now taking place through polycentric transnational networks, which do not depend on missionary logics. However, as Csordas (2015) rightly reminds us, these reorientations constitute the outline of a "global geography of the Spirit."

In our research, we have chosen to favour the concept of transnationalization, which we felt was more relevant than that of globalization. In the early 1990s, in a pioneering work, Glick-Schiller, Basch and Szanton Blanc (1993: 6) proposed the notion of transnationalism as a new analytical field for understanding migration, "to emphasize that many immigrants today build social fields that cross geographic, cultural, and political borders." A few years later, Hannerz (1996: 6) stressed the importance of using the term "transnational" instead of the "prodigious" use of the term "globalization" to refer to any process that crosses national borders, a term "in a way more humble, and often a more adequate label for phenomena which can be of quite variable scale and distribution, even when they do share the characteristic of not being contained within a state." 7The same unease with the use of this term has also been highlighted by Csordas (2009), for whom the influence of globalization is often thought to be fundamentally unidirectional, going from a globalizing centre to a periphery that would passively receive the global flows.

However, the studies we have carried out on the transnationalization of Afro-Atlantic religions show a reality that is diametrically opposed to this type of dynamic. For the most part, these are practices that generate multidirectional flows of people, goods and religious values. These flows-and the networks they generate-can change, their intensity can increase or decline, their scope of action can widen or narrow according to the various rearrangements of the religious modalities involved. Thus, while the term "transnationalism" occupies a central place in the Anglo-Saxon literature, we have preferred to replace it with the notion of "transnationalization" which allows us to emphasize the processes instead of underlining a sort of intrinsic quality of certain social phenomena.

"Cultural globalization" is obviously not a new phenomenon. We know that world history has seen other phases of globalization, driven by colonial expansion. Several authors have also emphasized the historical depth of these phenomena, since even among immigrants to the United States at the turn of the nineteenth century, transnational links between the society of origin and the host society could be observed. Transnationalism is therefore not a new phenomenon, but a "new perspective" (Portes, 2003: 874). The transnational approach-"the transnational lens"-help us to capture contemporary phenomena as the current "sense of simultaneity" [START_REF] Levitt | Conceptualizing Simultaneity: A Transnational Social Field Perspective on Society[END_REF] of inhabited worlds or the feeling of living in a "global world." This "sense of simultaneity" is not only due to what has been called the "death of distance" which characterized our pre-Covid era, but also-and above all-to the rise of social networks, among them Facebook and Instagram, and of new communication tools as WhatsApp, which enormously facilitate contacts between initiates of different nationalities with the help of Google translator. The multiplication of these global interconnections makes it possible for the initiates in the Orisha religion to develop the feeling of living in shared worlds that are being built on both sides of the Atlantic.

This new perception brings in several changes in Afro-Atlantic religious practice.

Firstly, the multiplication of links between the various centres preserving African traditions, which are not only in Africa, but also in the Americas. Today different traditions of the Orisha religion are present in the Brazilian religious field, confronting Brazilian Candomblé with ritual practices from Cuba and Nigeria. This has contributed to the creation of an "ecumene," to use the notion proposed by [START_REF] Hannerz | Notes on the Global Ecumene[END_REF], for whom the Greek term oikoumene designates "a region of cultural interaction and exchange," a notion which also evokes that of the black Atlantic, elaborated by [START_REF] Gilroy | The Black Atlantic. Modernity and Double Consciousness[END_REF]. This transnational space, this "Yoruba ecumene"-Yoruba culture being the origin claimed by many Afro-Atlantic religions-is then a co-construction not only between Africa and the Americas, but also between the various centres producing "traditional" discourses and practices on the American continent, as Salvador de Bahia in Brazil, or Havana and Matanzas in Cuba.

Moreover, the study of Afro-Atlantic religions questions the very opposition between the bond to a territory (the "local" or the "national"), which would produce "pure" and "authentic" cultures, and the deterritorialization associated with the transnational, which would, in turn, put forward "hybrid" or "creolized" cultures [START_REF] Werbner | Introduction: the dialectics of cultural hybridity[END_REF]. In reality, at least for these religions, transnationalism does not prevent an essentialist discourse, in which culture, despite its obvious transformation and adaptation, is still thought of as "pure" and "traditional." Transnationalization does not necessarily lead to phenomena of hybridization or creolization, as it is shown in Candomblé's ritual re-Africanization (Capone, 2016a).

Transnationalized practices often reinvest their localities of origin by reviving cultural traditions, which are promoted to the rank of vectors of universality, as in the case of the Ifá cult, a divinatory system organized around two "national" traditions, the Nigerian and the Cuban. I will therefore focus my analysis on the new configurations generated by the unprecedented encounter between different models of tradition within the Orisha religion.

Before presenting some ethnographic data, I propose to examine two issues that are at the very heart of my research on transnational fields: the weight of the nation, which is not automatically erased by transnational processes, and the possibility of being transnational "without moving."

The weight of nation

In our research, we have worked with two different concepts: "transnationalization" and "translocalization." However, the terms "translocal" and "transnational" do not refer to the same processes or the same scales of analysis. While it may be appropriate in certain situations, the term "translocal" evacuates the weight of the nation as a receptor or exporter of religious practices, which are often conceived as part of a national cultural heritage, as for Afro-Cuban religions that bring with them a strong nationalist component. Indeed, the weight of "national imaginaries" makes Afro-Brazilian and Afro-Cuban religions-based on traditions emphasizing a logic of dual affiliation-both endogenous and exogenous religious practices. They are of African origin but are also-and above all-Brazilian or Cuban. These "national" identities are inscribed at the very heart of religious identities. For example, when a Candomblé follower is initiated in Brazil into the Ifá cult according to Cuban tradition, he will learn "Cuban style" Spanish in order to be able to read Cuban Ifá treatises or other sacred texts, he will adopt Cuban prayers and invocations, he will seek contacts with Cuban or Cuban-American initiates, while negotiating the place of this new religious identity alongside his previous ritual affiliations. This also involves the use of a national language that becomes the ritual language in processes of religious transnationalization. For example, the Orisha manifestation in the United States is often subject to "truth regimes" originated from its context of origin, namely Cuba, from where the Orisha worship arrived in the 1960s. The use of Cuban-style Spanish by the embodied Orisha then becomes the proof of the veracity of the initiate's possession.

A "real" Orisha will then speak Spanish in Lucumí ceremonies in Miami or New York, as a "real" spirit will speak Portuguese in Umbanda rituals in Paris or Rome. In the implantation of Afro-Brazilian and Afro-Cuban religions in the United States, Argentina, Uruguay or Europe, the issue of language thus acquires a "sacred" character.

Communicating in Portunhol, a mixture of Portuguese and Spanish, has become the sign of an "authentic" possession in the countries of the Latin American Southern Cone [START_REF] Frigerio | L'umbanda et le batuque dans le Cône Sud: du flux religieux transfrontalier au champ social transnational[END_REF].

However, the mastery of ritual language is also accompanied by the production of multi-layered identities, since in these religions the notion of conversion does not apply. Indeed, when a practitioner undergoes new initiations, he does not abandon his previous religious practices but rather accumulates them with the new ones. Sometimes, these stratified religious identities also generate ritual changes that can lead to an "ethnicization" or even "racialization" of Afro-Atlantic religions, as in the North American case of Oyotunji Village [START_REF] Clarke | Mapping Yorùbá Networks: Power and Agency in the Making of Transnational Communities[END_REF]Capone, 2005). This revives the tensions between an idealized African origin and a diversified American religious practice that has long included whites as well in countries that are still suffering of structural racism and social inequalities.

Therefore, the ritual rearrangements within the Orisha religion do not abolish the different national scripts, but remain indebted to the national histories of each country [START_REF] Frigerio | Guerre sainte dans le Cône sud latino-américain : pentecôtistes versus umbandistes[END_REF].

The term "transnational" should not then automatically refer to the erasure of the nation-state, since it often represents a redeployment of its prerogatives. Today, social actors can also embody their "national culture" or a certain "national ethos" in transnational settings. As [START_REF] Clifford | Routes: Travel and Translation in the Late Twentieth Century[END_REF] reminds us, translocal phenomena are always embedded in particular geographies and histories, which are very often those of nationstates. The enduring presence of the "national" in the "transnational" is certainly one of the most stimulating challenges in transnational studies.

Transnational (im)mobilities or how to be transnational "without moving"

The second point I would like to emphasize is that religious transnationalization is not necessarily linked to migration. The thematic of (im)mobility is calling increasing attention in the field of migrations [START_REF] Easthope | Fixed identities in a mobile world? The relationship between mobility, place, and identity[END_REF][START_REF] Moret | Mobility Capital: Somali migrants' trajectories of (im)mobilities and the negotiation of social inequalities across borders[END_REF][START_REF] Schewel | Understand Immobility: Moving beyond the mobility bias in migration studies[END_REF], highlighting the mutual constitutive relationship between mobility and immobility (Kael, 2021;[START_REF] Rocha | New Perspectives on the Globalization of the Brazilian Religious Field[END_REF]. In a previous work (Capone, 2010b), I developed a reflection on mobility and immobility in transnational settings.

We know that it is in the field of international migration studies that the theoretical frameworks of transnationalism have been elaborated. This new approach emphasizes the links between "here" and "there," staging a "circulatory territory," to use the expression immortalized by [START_REF] Tarrius | Les nouveaux cosmopolitismes. Mobilités, identités, territoires[END_REF], in which migrants develop an awareness of belonging to two worlds at once, to their land of origin as well as to their host country. In the early 1990s, Basch, Glick-Schiller and Szanton Blanc (1994: 6) proposed the term "transmigrant" to refer to the multiplicity of relationships that immigrants maintain with their country of origin.

However, while it is true that the activities of religious "transmigrants"-namely the pais and mães de santo (the cult leaders of Afro-Brazilian religions) who cross borders to take care of their foreign initiates-have been fundamental in the process of implantation of Afro-Atlantic religions, our research has shown that the processes of expansion of these religious practices can also take place without a strong presence of immigrants and beyond any missionary enterprise. In reality, movement is not a prerequisite for any transnational action. While some migrants periodically cross borders, there are also a large number of individuals whose lives are deeply embedded in the host society. Nonetheless, these immigrants are still inscribed in networks that link them-through flows of people, goods or information-to their country of origin. They may not move physically, but they live their lives in a context that has become "transnational," imagining themselves as members of a group that is constituted across space. Individuals who do not move maintain social relations across borders through various forms of communication. As Peggy [START_REF] Levitt | Redefining the Boundaries of Belonging: The Institutional Character of Transnational Religious Life[END_REF]Nina Glick-Schiller (2004: 1009) point out, the actions and identities of those with weak connections to their society of origin are not less influenced by transnational dynamics. Hence, migrant communities are not the only ones affected by religious transnationalization. Transnational networks can also be formed while remaining at home, thanks mainly to the new means of communication, such as Internet. The concept of (im)mobility can then be a tool for thinking about how some people are able to fully explore transnational networks, while others continue to practice their faith on local basis in a world where digital technologies have become crucial to sociality and religious identity construction. [START_REF] Salazar | The Power of Imaginaries in Transnational Mobilities. Identities[END_REF] has stressed the centrality of imaginaries in providing the cultural material to be used for the creation of translocal connections. Initiates who navigate in transnational networks accumulate what she calls "cosmobility capital," resources, knowledge and abilities that facilitate social, geographical, and religious mobility. This issue has accelerated with the Covid-19 pandemic, but previous works have demonstrated that mobility is a contested ideological construct involving much more than mere movement. It will be then interesting to explore the intersection of (im)mobility and new configuration of belonging in a digital world.

Transolorisha: agency and power in transnational networks

In a book in preparation (Capone, forthcoming) on the religious diasporic connections between Brazil and Nigeria, I propose the term "transolorisha," where olorisha designates an initiate in the Orisha religion (o + ní + òrìsà = the one who possesses the Orisha), for the initiates circulating between American and African countries.8 A transolorisha is thus an initiate in one of the variants of the Orisha religion in Brazil or elsewhere, who travels to Nigeria (or to another centre of Orisha tradition in the Americas) to carry out new initiations that will integrate him or her into new networks of social and spiritual relationships. The term "transolorisha" captures this relationship to an elsewhere that does not necessarily imply a continuous movement between two localities.

Transolorishas are not migrants but travellers between two cultures, always in search of a more direct contact with the "source" of the "African tradition" in a space that is being constructed on both sides of the Atlantic.

In our research, we conceived it as a space of relationships, as a space of circulation of people, objects, practices, symbols and ideas. In other words, beyond the national and regional contexts in which these practices are deployed, there is a broader space that encompasses the multiple connections woven by the religious actors, and where a particular perception of belonging is experienced, the "feeling of translocality" described by [START_REF] Falzon | Cosmopolitan Connections. The Sindhi Diaspora, 1860-2000[END_REF] in his ethnography of the transnational network of Hindu traders, or the "sense of belonging," proposed by Peggy Levitt and Nina Glick-Schiller (2004). But this feeling can also be produced on a virtual level. Indeed, the Internet, and especially the social networks, have become a complementary but indispensable research field, which helps us to reconstruct the networks of individuals and groups interconnected by virtual links (Capone, 1999b).

While the transnationalization of Afro-Atlantic religions is made possible by a process of interaction between ritual specialists and individuals of different nationalities, who move back and forth between different countries, one can also be transnational without moving. Indeed, many priests and priestesses of Afro-Atlantic religions do not necessarily travel, but have to handle religious knowledge and ritual protocols that can be different from those of their local context. Without physically traveling, they can remain connected to other places (through visits, telephone, especially WhatsApp, and the social networks), sharing cultural codes, ways of thinking or implicit references that allow for "conversation" [START_REF] Comaroff | Of Revelation and Revolution: Christianity, colonialism, and consciousness in South Africa[END_REF] with their "brothers in religion," in Nigeria or elsewhere.

Research on religious transnationalization is thus not only about migration dynamics, about the movement of individuals, but also about the consciousness of belonging to a "globalized" world and the imaginary of being part of two worlds at once. The anthropologist can even do "multi-sited research" [START_REF] Marcus | Ethnography in/of the World System: The Emergence of Multi-sited Ethnography[END_REF] while remaining in one place, by analysing, for example, the connections, exchanges, borrowings, and tensions between different belief systems or between different regional traditions. This is the case, for example, of my fieldwork in Rio de Janeiro, where I work with specialists in Ifá divination, who follow several traditions: the Nigerian, the Cuban, but also, more recently, the Brazilian one, a sort of synthesis of these two "national" traditions.

Brazilians undergo initiations with Cubans and Nigerians, and must master, or at least try to master, the cultural codes and rituals of each tradition. This mastery, more or less accomplished, plays a central role in the processes we study. Practitioners of Afro-Atlantic religions can then develop a feeling of being members of transnational ritual networks, "inhabitants" of a multi-territorialized space of relations, without necessarily "moving." Indeed, initiates always emphasize their identification with ritual lineages, stressing the links with the "land of roots," a relation that draw new strength from the gigantic weave of other virtual links connecting initiates on the Net. However, to master this plurality of ritual idioms imply also a form of reflexivity about one's own identity, redefining the boundaries of the religious group, the different "geographies of power" that link it to other religious traditions, as well as the possibilities of a dialogical relationship with other national traditions. The relationship between agency and power is at the heart of religious transnationalization processes. The implications of these processes of reconfiguration of "traditional" power and prestige within the transnationalized religious practices are, however, issues that have not yet received the attention they deserve in transnationalism studies. Some examples from my research on Ifá transnationalization in Brazil will show how the transnational context can profoundly alter the delicate balance between Afro-Atlantic religions, disrupting their power relations and hierarchical structures.

Power, Gender and Possession

One of the most significant changes in the Afro-Atlantic religious field has been the diffusion, since the 1960s-1970s, of religious practices of African origin beyond ethnic and national borders, with their implantation in new countries that did not necessarily have a long religious tradition comparable to those of the countries that saw the birth of Afro-Atlantic religions (Brazil, Cuba and Haiti). In recent decades, these religions have also profoundly modified their image, occupying the public space in unprecedented ways and claiming their cultural specificity. From being "black and poor" religions, they have been transformed, especially in Brazil, into religions whose practitioners today come from different social backgrounds, including foreigners, who import these religions into their own countries. Today, transnational circulation has thus led to an unprecedented confrontation between different models of tradition, that must deal with a deeply stratified and fragmented universe. Indeed, while certain religious modalities, such as Ifá cult and Brazilian Candomblé, all refer to the same Yoruba origin, this "pan-Yoruba" identity is never unique. On the contrary, it is characterized by its multiplicity, by different national religious identities and by their interaction, which is often conflictual.

In another works (Capone, 2014(Capone, , 2016b)), I have analysed the religious transnationalization brought about by the reintroduction of the Ifá cult in Brazilian Candomblé temples. I will just resume some points that disclose the impact that these processes are having on the Afro-Atlantic religious field through structural changes in the religious hierarchy. Among the religious practices of Yoruba origin, the Ifá divination system occupies a central position. The spread of the Ifá cult in the world and its knowledge inscribed in the corpus of odù, the divinatory "signs," offers for the first time to the initiates into Afro-Brazilian religions the outline of a sacred Book that can be the basis of their religious practice. This special position allows the babalawo (the priest of Ifá divination) to develop hegemonic aspirations over the other local variants of the Orisha religion.

Recently, some cult leaders have even begun to claim a World Religion status for what they call "Ifaism"-a new religious variant based on the worship of Ifá/Orunmilá and his sacred scriptures, and which is often presented as a kind of monotheism. In Brazil, the Ifá cult-which is part of the Orisha religion since it is placed under the tutelage of the god Orunmilá-was revitalized in the 1980s, after having fallen into oblivion following the death of the allegedly last Brazilian babalawo in the 1940s. In a previous work (Capone, 2010a[START_REF] Capone | La quête de l'Afrique dans le candomblé. Pouvoir et tradition au Brésil[END_REF]), I showed how Yoruba language courses, which were quickly transformed into divination classes according to the Ifá system, prepared for the arrival of Nigerian babalawo in Brazil and, from the early 1990s onwards, of Cuban babalawo who defend their own Ifá tradition. The Candomblé initiates seek, in fact, to tirelessly deepen their knowledge of Yoruba culture, complaining that "the elders" did not transmit all the knowledge they had to the new generations. It is this unfinished transmission of sacred knowledge that has led to the transnationalization of other Afro-Atlantic religious modalities and their implantation in Brazil.

The issues of gender and possession are at the very heart of the rearrangements brought about by this religious transnationalization. In my work,9 I have shown how the presence of Cuban babalawo in Rio de Janeiro allowed the initiates of Candomblé to become familiar with new models of tradition, where men, and in particular the ogan-a ritual office reserved in Candomblé for heterosexual men who do not enter into a trancefound in the initiation into the Ifá priesthood a new way of access to the highest positions in the religious hierarchy.

In fact, the babalawo priesthood, according to Cuban tradition, is restricted to heterosexual men who must not be possessed by the gods. Women can be initiated into the Ifá cult, but they occupy a lower place in the hierarchy, becoming iyapetebí, the babalawo's assistant who cannot perform divination with the opelè (the chain of divination that is an attribute of the babalawo). The only possible consecration for women in the Cuban Ifá cult is therefore the ceremony of Kofá or Ikofá, which corresponds to the first level of initiation for men, called

Awo Fakan, a level to which homosexuals are confined.

On the contrary, in Brazilian candomblé, religious power is concentrated in the hands of women (mães de santo) and of men (pais de santo) which are often homosexuals. All of them embody their deities in ritual ceremonies. The candidates for initiation who do not enter into a trance-i.e., the ogan-occupy high positions in the hierarchy of the Candomblé temples, but they will always remain subject to the authority of their initiator. Thus, an ogan cannot initiate other people, because one of the necessary conditions for the reproduction of religious lineages in Candomblé is the direct experience of trance and the development of mediumnity, that is to say, the capacity to embody the deity of which the initiate is considered the spiritual "son" (filho de santo). Without experiencing and mastering the trance of the gods, no one can initiate a novice in this religion.

In this encounter between Afro-Cuban and Afro-Brazilian traditions, linked by the same claims of a Yoruba origin, the main points of tension between the babalawo and the Candomblé initiates are then the importance given in Candomblé to feminine power, the role played by homosexuals, and the centrality of possession. In fact, in Cuban Santería, as in Nigeria, possession is not a prerequisite and an indispensable condition for initiation, as it is in Candomblé for the filhos de santo. The ogan thus become the best candidates for initiation into the Ifá cult, since they escape any suspicion of simulation that the Cuban babalawo put on Candomblé initiates, accusing them of not embodying gods, but spirits of the dead who can therefore be chased away. The centrality of possession as the basis of ritual practice in Candomblé is therefore deeply questioned by the initiates in the Ifá cult.

Moreover, the integration of Ifá priesthood into Candomblé also entails a real inversion in the religious hierarchy, since, according to Cuban tradition, the babalawo is considered "superior" to the olorisha, as Orunmilá (the god of divination) is "superior" to the other Orisha. Thus, a babalawo, who has just been initiated, will automatically become the "elder" of an olorisha with thirty years of initiation. This openly goes against the hierarchical organization of Candomblé, which is based on a strict principle of seniority. In the last years, this opposition between the Orisha worship and the Ifá cult is turning the universe of Afro-Atlantic religions upside down, triggering a struggle for religious supremacy in which the Nigerian babalawo play a fundamental role [START_REF] Capone | Ifá reconquiert le monde ou les défis d'une « nation yoruba imaginée[END_REF]. By emphasizing the omniscience of Orunmilá, the Witness of Destiny, the babalawo use their knowledge, supposedly more "rational," to impose their supremacy over the initiates in the Orisha religion, sending the olorisha back to a form of knowledge supposedly inferior to the imo jinlè, "the deep truth" conveyed by Ifá. This vision has triggered a sharp opposition between initiates in the Orisha cult and babalawo, both in Brazil and in Nigeria.

The babalawo defend the superiority of their cult, affirming that all the Orisha acknowledged it by being initiated into the Ifá cult. Yet the olorisha are far from accepting the tutelage of the babalawo and a controversy has broken out in 2019 and still continues on social networks, pitting traditional Orisha families and babalawo associations in Nigeria against each other. This controversy has also direct effects on the "diaspora," fuelling debates about which tradition is closer to "African roots."

The multiple differences between the Orisha worship in Brazil, Cuba and Nigeria are always interpreted by the babalawo, as being the consequence of a fundamental loss of religious knowledge, which would have produced this gap between matrices of meaning.

For them, the differences between ritual practices in this transnational space do not call into question the strength of Yoruba culture, but is the product of "holes" in the African collective memory [START_REF] Bastide | Les religions africaines au Brésil. Contribution à une sociologie des interpénétrations de civilisation[END_REF] that can be filled today by reinjecting cultural content and practices that had fallen into oblivion in Brazil or elsewhere (Capone, 2007).

Exchanges between "sister religions," such as Brazilian Candomblé and Cuban Santería, thus aim at re-establishing a common belief system, in which elements of different Afro-Atlantic religions, all claiming the same Yoruba origin, are combined in different ways.

Productive misunderstandings and transnational imaginaries

The irruption of Ifá into Candomblé provides new forms of legitimization of ritual practice. Ifá is thus becoming the major place of production of meaning and articulation of religious knowledge, giving rise to new transnational imaginaries (Appadurai, 1996). These imaginaries are constantly being questioned, since they modify, as we have seen, the relations between genders in the division of religious work, transforming them into a real ground of negotiation between "local" and "global" cultures.

Nevertheless, the inscription in the local is not only done through the congruence of imported practices or a tuning in with pre-existing cognitive frameworks. In these "diasporic conversations," the discourse displayed does not necessarily have to be always coherent and in congruence with the beliefs and values of the targeted individuals. In the transnationalization of Ifá cult in Brazil, I also disclosed the importance of fuzzy and ambiguous discourses, since they elicit a multiplicity of interpretations and appropriations.

Adaptation strategies can therefore play as much on what is said as on what is not said, as much on negotiation at the ritual level as on misunderstandings, which can be more or less productive [START_REF] Sahlins | Islands of History[END_REF].

Misunderstanding thus becomes a form of interpersonal and intercultural strategy that is likely to prepare, produce and consent the encounter between different belief systems (Capone, 2011). Whenever a consensus has to be established in this "dialogue" between religious actors from different traditions, one will resort to what Bourdieu (2001: 64) called a "neutralized language" and Galison (1997) calls a "trading zone." Consensual signifiers will then be mobilized-terms, images or objects whose simplicity seems to be agreed upon-but behind which each actor will put a different meaning. It is in these "zones of awkward engagement" [START_REF] Tsing | Friction. An Ethnography of Global Connection[END_REF] that the dynamics of cultural interaction at the global level become intelligible. It is in the tensions-Anne Tsing would call them "frictions"generated by the encounter between different modalities of worship, all of which are part of the same "Yoruba tradition," that the "black Atlantic" is configured as a community of sensibilities and destinies. Each friction, each clash in the cultural encounter reconfigures this imagined community (Anderson, 1983).

The confrontation between multiple visions of the tradition and the polemics that ensue thus constitute key moments that produce configurations, always unstable, of the Orisha religion at the transnational level. These new configurations are made possible by the work of misunderstanding (Capone, 2011) or by a "productive confusion" which, according to Tsing (2005: 247), is "the most creative and effective form" of cultural collaboration. However, if misunderstanding lays the foundations for establishing a dialogue, it also makes it possible to manage the boundaries between belief systems that are at once close and opposed. Indeed, the Orisha religion, despite its unifying discourse, mobilizes universes that are not always compatible, as we saw in the opposition between Ifá cult and Candomblé. The transnational social space, connecting the African motherland to the diaspora, as well as the various centres of the African tradition on American soil, operates an entanglement of imagined worlds that poses in new terms the question of boundaries between modalities of worship. This transnational social space is neither egalitarian nor homogeneous, but the result of power structures and internal boundaries-historical, political and ritual-that cut across the networks of Orisha practitioners. Ritual lineages thus constitute social and symbolic boundaries that must be constantly renegotiated when new religious practices are grafted onto pre-existing ones. Conflict is then often the result of the disruption of the boundaries between these "communities;" it is the result of tensions or "frictions" between the different cults involved. This encounter between traditions, claiming the same Yoruba origin, proves that religious globalization cannot be thought of as a homogenizing factor.

On the contrary, despite the appeals to a transnational Yoruba imaginary, elevated to the rank of a matrix of meaning by incessant references to the Ifá sacred literature preserved by the babalawo, the Orisha religion today constitutes a conflictual space, built around a structural tension between homogenization and heterogeneity of cultural practices.

Analyses of the processes of religious globalization must therefore take also into account the tensions, conflicts, adjustments and frictions between the religious systems involved, as well as the new responses they produce in this confrontation between a globalized Yoruba imaginary and localized Afro-American traditions. While some religious modalities, such as Ifá cult and Brazilian Candomblé, all refer to the same Yoruba origin, this identity is characterized by multiple national religious identities and by their interaction, which is often antagonistic. While there is a "sense of belonging" [START_REF] Levitt | Redefining the Boundaries of Belonging: The Institutional Character of Transnational Religious Life[END_REF] to an "imagined community," such as that of the practitioners of the Orisha religion, there is also an awareness of the sometimes insurmountable differences between the multiple national versions of the "Yoruba tradition."

In the opposition between variants of the Orisha tradition, all considered "traditional" as Ifá cult and Candomblé, it is around the multiple notion of tradition that ritual negotiation takes place. Religious knowledge is therefore a scarce resource, since the discourse of the members of the Candomblé tirelessly emphasizes the preservation of an ancestral cultural and ritual heritage that is incomplete because of the unfinished transmission of ritual knowledge from initiator to initiate. In the Orisha religion, knowledge is the basis of religious power and, for this very reason, must remain limited.

Moreover, the effort to recollect ancestral knowledge must be inscribed in a territory; it must be "localized." Any "deterritorialization" must be followed by a new "reterritorialization," whether real or symbolic (Capone, 2004a). However, the old divisions between regional or national traditions are never erased, resurfacing in the processes of memorialization of a tradition that wants to be millennial. What has changed today is that, instead of defining cultural "niches" or territories of resistance [START_REF] Bastide | Les religions africaines au Brésil. Contribution à une sociologie des interpénétrations de civilisation[END_REF], this geography of memory no longer has a single referent, but a complex network that points to the many traditional centres of the Orisha religion: Ilé-Ifè or Oyó in Nigeria, Havana or Matanzas in Cuba, Salvador de Bahia in Brazil or, more recently, Oyotunji Village in the United States. All these places delimit the perimeter of the same symbolic community which does not exclude conflict and segmentation as a form of religious reproduction. This is the main challenge facing the Orisha religion today, torn between the aspirations of becoming a universal religion and the constraints of its inscription in particular national histories.

Today these religions are generally called "Afro-Atlantic," a term which stresses their co-construction on both sides of the Atlantic Ocean.

The Yoruba are an ethnic group, concentrated mainly in Nigeria and Benin, which paid a high price to the trade of enslaved Africans and whose culture is at the origin of the formation of several Afro-Brazilian and Afro-Cuban religions. Today, a person initiated into these religions in the Americas can develop a sense of belonging to an ancient African religion and culture that is independent from her real ethnic origins. A white Brazilian can thus become a strenuous defender of the Yoruba culture, that becomes part of her identity construction through initiation.

This expression, which has become a powerful metaphor in the Afro-Atlantic religious field, refers to religious practices that worship the Orisha, i.e. the gods of Yoruba origin.

Cf. Capone (1999a[START_REF] Bourdieu | Langage et pouvoir symbolique[END_REF]-2002, 2004b, 2005, 2016a, forthcoming).

This project, directed by Kali Argyriadis, gathered 21 researchers from France, Mexico, Brazil, Colombia, Argentina, and Gabon, organized in 7 partner institutions: IRD, CNRS/University of Paris X and CEIFR/EHESS in France, CIESAS-Occidente and El Colegio de Jalisco in Mexico, Omar Bongo University in Gabon, and the Federal University of Rio Grande do Sul in Brazil.

Cf. Capone (2004a), Bava and Capone (2010), Argyriadis and Capone (2011), Argyriadis, Capone, De la Torre and Mary (2012),Capone and Salzbruun (2019).

In another work(Capone, 2004a), I analyzed the relationship between the notions of globalization and transnationalization.[START_REF] Ong | Flexible Citizenship. The Cultural Logics of Transnationality[END_REF] offers an interesting analysis of the "transnationalism-globalization" dialectic that opens up some challenging avenues.

A previous group of research, linked to the Mexican team of the RELITRANS project, suggested the term transaborisha (or transaboricha to respect the Spanish orthograph) to designate the practitioners of Yoruba and Yoruba-derived religions involved in transnational networks. However, this term refers to Orisha worshippers (the Yoruba verb bo meaning "to worship") while the majority of people involved in transnational networks are already initiated into several Yoruba-derived religious modalities in the Americas. The term "transolorisha" tries to capture the layering of successive initiations, performed in Africa and the Americas, producing new religious identities.

See, for example, Capone (2014, 2016b).

Bibliography


Julien Villemonteix

Emmanuel Vazquez
email: emmanuel.vazquez@supelec.fr.

Eric Walter

Bayesian optimization for parameter identification on a small simulation budget

Keywords: Gaussian processes, global optimization, Kriging

Bayesian optimization uses a probabilistic model of the objective function to guide the search for the optimum. It is particularly interesting for the optimization of expensive-to-evaluate functions. For the last decade, it has been increasingly used for industrial optimization problems and especially for numerical design involving complex computer simulations. We feel that Bayesian optimization should be considered with attention by anyone who has to identify the parameters of a model based on a very limited number of model simulations because of model complexity. In this paper, we wish to describe, as simply as possible, how Bayesian optimization can be used in parameter identification and to present a new application. We concentrate on two algorithms, namely EGO (for Efficient Global Optimization) and IAGO (for Informational Approach to Global Optimization), and describe how they can be used for parameter identification when the budget for evaluating the cost function is severely limited. Some open questions that must be addressed for theoretical and practical reasons are indicated.

INTRODUCTION

The identification of a parametric model (or parameter estimation) is usually carried out by minimizing some cost function f (x) that quantifies the difference between experimental data and results of model simulation, where x ∈ X is the parameter vector of the model. When this cannot be done analytically, minimizing f by conventional iterative methods, possibly combined with multistart to try to escape local minima, generally requires many model simulations. We assume in this paper that the number of model simulations (or function evaluations) effectively achievable is severely limited by either time or cost.

It then becomes essential to look for optimization methods that use the information available as efficiently as possible. Such methods often use an approximation based on the results of past evaluations as a cheap proxy for the function to be optimized. We shall refer to this proxy as a surrogate approximation to avoid confusion with the parametric model.

During the last decade, surrogate approximations relying on a probabilistic model of the function to be optimized have received particular attention [START_REF] Huang | Experimental Planning and Sequential Kriging Optimization Using Variable Fidelity Data[END_REF][START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF], Villemonteix et al., 2008b]. The field of optimization techniques that rely on such models is known as Bayesian Optimization [START_REF] Zilinskas | A review of statistical models for global optimization[END_REF]. The interest of a Bayesian approach to the optimization of expensive-to-evaluate functions has already been discussed extensively (especially in [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]) and confirmed by many applications to industrial design problems (see, e.g., [START_REF] Huang | Design and analysis of variable fidelity experimentation applied to engine valve heat treatment process design[END_REF], Villemonteix et al. [2008a]). The context of restricted evaluation budget indeed makes most algorithms inefficient. Since the mere simulation of complex knowledge-based parametric models often requires an heavy computational effort and the usual iterative algorithms for nonlinear parameter estimation rely on a large number of such sim-ulations, the potential of Bayesian optimization in parameter estimation must be investigated.

Our first results in this context were presented in [START_REF] Villemonteix | Identification of expensive-to-simulate parametric models using kriging and stepwise uncertainty reduction[END_REF], but practical details were reduced to a minimum, and no real-life application was proposed. In this paper, we wish to describe, as simply as possible, how Bayesian optimization can be used in parameter identification and to present an application. We shall concentrate on two algorithms, both based on Gaussian processes and Kriging, namely the reference algorithm EGO (for Efficient Global Optimization [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]) and a very recent alternative IAGO (for Informational Approach to Global Optimization [Villemonteix et al., 2008a,b]).

In Section 2, we recall the principles behind EGO and IAGO. We give some details about these algorithms in Section 3, present an application in Section 4, and finally discuss some open questions in Section 5.

GAUSSIAN PROCESSES FOR OPTIMIZATION

Bayesian optimization is based on two main ideas. The first one is to model the cost function f (•) by a random process F(•), here assumed to be Gaussian with mean function m(•) and covariance function k(•, •) (in what follows, we shall simply call them mean and covariance). This means in particular that for S n = {x 1 , ..., x n } ⊂ X a set of evaluation points, the vector of evaluation results f n = [ f (x 1 ), ..., f (x n )] T is viewed as a realization of the random vector F n = [F(x 1 ), ..., F(x n )] T . The second idea is to choose the evaluation points iteratively based on what has been learned through the previous evaluation. If n evaluations have been performed at the points in S n , the (n + 1)-th point is chosen by optimizing a sampling criterion J(x, S n , f n , F) that measures the interest of an additional eval-

J(x, S n , f n , F). ( 1
)
In summary, Bayesian optimization replaces the optimization of an expensive-to-evaluate function by a series of optimizations of a cheap criterion. This criterion quantifies the interest of any additional evaluation, and reflects our concern for a careful spending of the evaluation budget.

In this section, we shall discuss how to build a surrogate approximation using Kriging, before giving some details about two possible sampling criteria build from this surrogate approximation. But first, we shall illustrate what we are trying to achieve. Assume that some surrogate approximation f (x) of f (x) for all x ∈ X has been built. A simple sampling criterion would then be the minimization of f (x). The optimization algorithm that ensues might converge towards a local optimum (cf. Fig. 1) and is completely dependent on the initial sampling scheme.

To make search more global, one should take into account the uncertainty associated with the surrogate approximation, and this is exactly what Kriging allows us to do.

An introduction to Kriging

Assume that the mean m of F is a finite linear combination of known functions, which can be written as m(•) = β β β T p(•), with p a vector of known functions, and β β β a vector of fixed but unknown coefficients. The theory of Kriging then addresses the construction, for all x ∈ X, of the best unbiased linear predictor (BLUP) F(x) of F based on the random vector F n . As a linear predictor, F can be written as F(x) = λ λ λ(x) T F n .

(2) Using a Lagrangian formulation, one can show that, for the predictor to be a BLUP, the vector of Kriging coefficients λ λ λ(x) must satisfy

K P P T 0 λ λ λ(x) µ µ µ(x) = k(x) p(x) , (3)
with K = (k(x i , x j )) , (i, j) ∈ 1, n 2 the covariance matrix of the evaluation results, k(x) = [k(x 1 , x), . . . , k(x n , x)] T the vector of covariances between F(x) and F n ,

P =    p(x 1 ) T . . . p(x n ) T    .
the regression matrix, and µ µ µ(x) a vector of Lagrange coefficients.

Given the covariance of F, the Kriging coefficients at x can thus be computed without evaluating f (x), along with the variance of the prediction error

σ2 (x) = E F(x) -F(x) 2 = k(x, x) -λ λ λ(x) T k(x) -p(x) T µ µ µ(x) , (4
) Usually, however, the covariance of F is not known a priori. It may then be chosen among a parametric family (for example, the Matérn covariance class [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF]), with the parameters estimated by maximum likelihood (we shall discuss this point in Section 5). Given the evaluation results f n , the prediction of f (x) becomes f (x) = λ λ λ(x) T f n which can be used as a surrogate approximation for f , while σ2 (x) gives us an explicit characterisation of the uncertainty associated with this prediction. The conditional distribution of F(x) is indeed Gaussian with mean f (x) and variance σ2 (x). Fig. 2(a) presents an example of prediction by Kriging. In what follows we describe two sampling criteria that use Kriging to balance local and global searches.

Maximizing the expected improvement

Sampling at a maximizer of the expected improvement (EI) was initially proposed by [START_REF] Mockus | The application of Bayesian methods for seeking the extremum[END_REF]. This strategy has been the object of most publications in the field in the last few years [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF][START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]] and has been used for industrial applications (see, e.g., [START_REF] Huang | Experimental Planning and Sequential Kriging Optimization Using Variable Fidelity Data[END_REF]).

A trivial estimator of the minimum of f after n evaluations is M n = min x i ∈S n F(x i ). The EI strategy chooses as the next evaluation point x n+1 , a minimizer of E[min(M n , F(x))|F n ], the expected value of this estimator after the evaluation at x and given the previous evaluation results

F n = {F n = f n }. This amount to maximizing the expected improvement EI(x) = E[max(M n -F(x), 0)|F n ],
(5) which represents the average excursion of F(x) under the current estimate of the minimum. The EI has a closed-form expression [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF], which involves both the Kriging prediction f and the variance σ2 of its error EI

(x) = σ(x)Φ ′ (u) + uΦ(u), (6) with u = (x -m n )/ σ(x), m n = min x i ∈S n f (x i ) and Φ(•) the Gaussian cumulative density function.
As can be seen on Fig. 2(c), the EI criterion favors sampling where the prediction is small, but also where the uncertainty on this prediction is large.

Minimizing the conditional entropy of the global minimizers

Where EI concentrates on the global minimum, we proposed in Villemonteix et al. [2008b] to concentrate rather on the global minimizers, which is of particular importance in a context of parameter estimation. Instead of considering an estimator of the minimum, we estimate the probability distribution of the global minimizers. Let X d be a finite subset of X and denote by X * a minimizer of F over X d . The probability distribution P X * (•|F n ) = P(X * = •|F n ) of the random vector X * given all previous evaluation results can then be approximated using conditional simulations of F [Villemonteix et al., 2008b]. Conditional simulations are simulations of F that interpolate all available evaluation results (see, e.g., [START_REF] Chilès | Geostatistics, Modeling Spatial Uncertainty[END_REF] for details on how to generate them).

Assuming that X d is a suitable representation of X (we shall discuss this idea in Section 3.2), P X * (•|F n ) represents what has been learned (through evaluations) and assumed (through the Gaussian model) about the minimizers. The main idea is then to quantify the uncertainty left on the location of these minimizers by the entropy of P X * (•|F n ). The progress made in finding a solution to the optimization problem is thus summarized by a scalar.

We then suggest to sample where the largest uncertainty reduction is expected, by minimizing the conditional entropy of the global minimizers (CEM)

H n (x) = Z y∈R p F(x) (y|F n )H(X * |F n , F(x) = y)dy, ( 7
)
with p F(x) (•|F n ) the conditional distribution of the evaluation result F(x) at x (Gaussian with mean and variance obtained by Kriging) and

H(X * |F n , F(x) = y) = -∑ u∈X d P X * (u|F n , F(x) = y) log 2 (P X * (u|F n , F(x) = y)) (8) the entropy of P X * (•|F n , F(x) = y), the distribution of X * given
F n and {F(x) = y}. Practical details on the evaluation of ( 7) and particularly the approximation of p F(x) (•|F n ) are discussed in Villemonteix et al. [2008b].On Fig. 2(c) and 2(d), we can compare, on the same simple example, the EI and CEM sampling criteria. Their difference appears particularly clearly near the origin. EI samples at x = 0, clearly not the most informative choice with half of its neighborhood outside search space.

USING IAGO AND EGO

In the previous section, we have described briefly the principles of two sampling criteria for optimization. Both have been especially conceived for a careful spending of the evaluation budget. In this section, we shall discuss how to insert these criteria into optimization algorithms, and give some recommendations regarding their practical use.

From the sampling criterion to the optimization algorithm

We have seen in Section 2 that the principle of Bayesian optimization was iteratively to choose points at which the cost function should be evaluated, by optimizing a sampling criteria J. If we use, as here, a Gaussian model and Kriging to build the surrogate approximation, the resulting optimization algorithm looks like Algorithm 1. First, f has to be sampled on a small initial design in order to initialize the Kriging predictor, and possibly to allow an estimation of the covariance parameters. Then, the Kriging predictor is computed, and an additional point is selected to be the location of the next evaluation of f , by optimizing the sampling criterion J, based on this predictor. The covariance function may then be updated (for example by re-estimating its parameters), the model is re-computed, and the process of choosing new points continues until some stopping condition is met. The optimization algorithms based on EI Compute the Kriging prediction based on all available evaluation results

5:

Evaluate f at arg opt x∈X J(x, S n , f n , F). 6: end while and CEM have been named respectively EGO and IAGO. The advantages of IAGO over EGO will be discussed in Section 5. However, it must be stated right away that the computational complexity of IAGO is significantly higher than that of EGO (cf. Villemonteix et al. [2008b]). The complexity of computing EI is O(n 3 ), while for CEM it is O(n 2 N), with N the size of X d . In practice, IAGO should only be used for simulation times that are significantly larger than the time needed to minimize the CEM sampling criterion (a few minutes on a personal computer for problems such as the one described in Section 4). Fig. 3 presents an application of IAGO to the sine function already used for Fig. 2. After twelve iterations, the three global minimizers have been identified as shown by the conditional distribution (right-hand part of Fig. 3).

In the rest of Section 3, we shall discuss how to optimize EI and CEM in practice, and the stopping condition. The covariance choice will be discussed in Section 5, along with the initial design as we wish to insist on the difficulties still left to deal with in practice.

Optimization of the sampling criterion

Optimizing the sampling criterion is an important practical of Bayesian optimization, as there may be many local optima. However, one has to bear in mind that the evaluation of the sampling criterion does not require any evaluation of the cost function f , and that there is no need for exact optimization since the sampling criterion is only used to determine the next evaluation point. A small error will have little influence on the final estimation of the minimum of f and of the corresponding minimizers.

At the beginning of the optimization procedure, we have very little a priori on the potential location of the minimizers. However, as the number of evaluations increases, it becomes obvious that certain areas of search space do not have any interest for an additional evaluation as they stand no chance of containing a global minimizer. Therefore, we propose to use P X * (•|F n ) as an auxiliary distribution for the optimization of the sampling criterion. For IAGO, this does not require any additional computation as this auxiliary distribution is already computed. The idea is to sample a set of candidate points from P X * (•|F n ) and to compute exhaustively the values taken by the sampling criterion over this set. An evaluation is then performed at the most interesting point; P X * (•|F n ) is recomputed and the set of candidate points re-sampled.

Note that, to compute P X * (•|F n ), a finite sub-set X d of X is needed. This set should represent the support of the probability density of the global minimizer of F over X. If it stays fixed as the number of evaluation increases, the support of this density will dwindle, as well as the capacity of X d to describe this support. That is why we propose to re-sample X d after each new evaluation using P X * (•|F n ) (estimated for example with a Gaussian kernel estimator [START_REF] Chib | Understanding the Metropolis-Hastings algorithm[END_REF]).

In summary, we can use X d as the set of candidate points and re-sample it to improve the precision of the optimization of the sampling criterion, and to maintain a good representation of the support of the probability density of the minimizers of F over X. The resulting optimization algorithm is summarized by Algorithm 2 (with either EI or CEM as the sampling criterion).

Algorithm 2. Bayesian optimization based on Kriging and the sampling criterion J (for example EI or CEM), when the criterion is computed extensively over a set of candidate points re-sampled after each new evaluation.

1: Evaluate f on an initial design 2: Choose an initial grid X d 3: while the stopping condition is not satisfied, do as the stopping condition, with σ stop = 0.1. Conventions are as in Fig. 3.

Stopping condition

Usually a simple stopping condition is available, namely exhaustion of the evaluation budget. It is indeed very common in practice to have exhausted the evaluation budget while still having a very poor estimation of the minimizers. Other stopping conditions have been proposed for either EGO [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] or IAGO [Villemonteix et al., 2008b]. They may be used when no budget has been specified beforehand. In this paper, we propose a new stopping condition, which is easier to use in practice. Let F * be the minimum of F and let var(F * |F n ) be the variance of F * given all past evaluation results. This quantity represent the uncertainty left on the estimation of the minimum.

We then suggest to stop the algorithm when

var(F * )|F n < σ stop , (9)
where σ stop is a positive threshold to be chosen by the user to reflect the desired precision on the solution. The conditional distribution of F * (an example of which is presented on Fig. 2(a)) can be approximated along with this variance using conditional simulations. Fig. 4 shows the impact of the new stopping condition on the example of Fig. 3.

AN INDUSTRIAL APPLICATION

The tuning of a parametric controller based on a performance criterion quantifying deviation form some ideal response is formally equivalent to parameter identification. We had to optimize the 32 parameters of the control law of an electrical power steering device (EPSD). This was based on data generated by a Matlab/Simulink model of the device interacting with a model of the car on a driving test case characteristic of the typical use of the vehicle. The response of the car to this test case can be visualised in the torque -steering wheel angle plane (cf. Fig. 5). Its quality was quantified by a cost function f reflecting the distance between the response and the ideal one (as specified The dash-dotted line represents the solution found by the Nelder-Mead Simplex method after 300 iterations. The points represent the results of the 300 evaluations on the LHS Sample. Finally, the dotted line represents the result of the evaluation at the minimizer of a polynomial model build from the 300 evaluation results of the LHS sample.

by experts in the field). Given the simulation time (10 minutes) for the test case and industrial needs, the minimization of f had to be carried out in less than 100 evaluations.

We dealt with this problem using EGO (preferred here to IAGO because of the relatively short simulation time) and reestimated the parameters of a Matérn covariance after each evaluation, using maximum likelihood estimation. We compared our results with those of two other optimization approaches. The first was to use the Nelder-Mead Simplex, and the second was exhaustive computation over a Latin Hyper Square (LHS) sample [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF] with 300 points. EGO turned out to be largely superior to the other approaches (see Fig. 6). It found in 45 evaluations of the cost function a better solution than that obtained via LHS sampling in 300 evaluations. Using the Nelder-Mead simplex was clearly the least efficient of the three approaches.

DISCUSSION

Choosing an algorithm. The interest of a given optimization algorithm can be tested in many ways. If convergence rates can be derived for a general class of functions, it becomes easy to nx * || over 1000 sample paths as a function of the number of evaluations of f , for IAGO (bold line) EGO (dotted line) and a random search (with a uniform distribution, dash-dotted line), when confronted to a irregular Gaussian process indexed on [0, 1] 2 . The standard deviation of the estimation of the mean is always less than 0.02. assert the interest of the algorithm compared to those of its competitors for which convergence rates have been obtained. When this is not the case, one may resort to large sets of test functions (see, e.g., [START_REF] Floudas | Handbook of Test Problems in Local and Global Optimization[END_REF]). An opinion can then be forged out of the precision obtained in the location of the minimum and minimizers on each test function, as well as the number of evaluations required. The practical interest of the latter approach can actually be questioned. If the comparison is conducted seriously, no method should turn out to be better than all others on every test case. A natural question is then, how can we deduce from the tests conducted which method to use a priori?

In the context of Bayesian optimization, we think that a more satisfying approach can be used to compare algorithms. For example, to compare EGO and IAGO, it seems natural to use sample paths of the Gaussian model behind both algorithms. By doing so, we can obtain empirical convergence results such as the one presented on Fig. 7. To obtain this result, 1000 sample paths of a Gaussian process (with X = [0, 1] 2 ) were generated and IAGO and EGO were applied to each of them. After each iteration and for each sample path, ||x * nx * || was computed, where x * n is an estimator of x * , obtained as a value of the parameter vector associated with the cost mn = min(m n-1 , f (arg max

x∈X d P X * (x|F n ))).

The vector x *

n is thus the solution that would have been retained in practice with a budget of n evaluation. On Fig. 7, IAGO appears to be significantly quicker to converge than EGO. The drawbacks of the use of EGO are evidenced on Fig. 8, where IAGO and EGO are applied on one of the sample paths used to obtain the results of Fig. 7. It appears that the EI stays stuck on a local minimum. This result has been obtained on a single Gaussian process. In Villemonteix et al. [2008a], we noted that, if the test process is the same as the model underlying IAGO and EGO, IAGO seemed regularly to outperform EGO. This approach could clearly be extended to any type of global optimization method. We could, for example, create a set of test random processes, carefully chosen to represent several types of regularity or other characteristics. Choosing a covariance. The choice of a covariance is a central question for Kriging prediction and, by way of consequence, for Bayesian optimization. Generally, the covariance is chosen in a parametrized class, and its parameters are estimated from available data. The method used for this estimation depends on the field (variogram fitting in geostatistics, cross-validation in machine learning or maximum likelihood in statistics [Villemonteix et al., 2008b]), but in a context of small evaluation budget, these methods may not be applicable.

[ [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] propose to initialize the EGO algorithm by an LHS sample of size 10d, with d the dimension of search space X. By doing so, the algorithm gets information on the promising areas in search space, but most of all a first estimation of the parameters can be carried out. This idea is appealing if d is small, but as d increases, the size of search space grows exponentially, and it soon becomes very hard to estimate the parameters of the covariance. Besides, if we consider the available evaluation budget for the application in Section 4, the size of the initial sample would have exceeded the total evaluation budget.

It finally appears that classical approaches for tuning the covariance do not apply in our context 1 . Now, EGO or IAGO may perform very poorly if the covariance is inadequate (see, e.g., [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF] for examples). Some preliminary results, not presented here, indicate however that when choosing the covariance inside the Matérn covariance class, it is possible to ensure a satisfactory behavior of IAGO or EGO by limiting search space for the covariance parameters. There even seems to be fixed covariances that ensure a satisfactory behavior of these algorithms over a large class of functions.

For the example of Section 4, we estimated the covariance parameters using maximum likelihood over a very restricted search space. In future work, we feel that a Bayesian approach would be best suited here to guide the estimation of the parameters of the covariance.

1 Except when evaluations results are available from a problem close to the one considered. In such a case the parameters can be estimated directly.

CONCLUSIONS

The main goal of this paper was to present EGO and IAGO, two Bayesian optimisation algorithms dedicated to problems with very restricted simulation budgets to the parameter identification community. We described the principles shared by the two algorithms and discussed their use in practice. The interest of Bayesian optimization for parameter identification was evidenced by an industrial application.

Comparison using the Gaussian process model behind EGO and IAGO indicates a clear superiority for IAGO. However, its computational complexity restricts its potential applications to the optimization of functions that require a large amount of computer time to be evaluated.

Figure 2 .

 2 Figure 2. (a) Example of prediction by Kriging (bold line) based on two evaluation results (circles) of a supposedly unknown function (dotted line). The dash-dotted lines delimit 95% intervals for this prediction, and the thin line is the conditional distribution of the minimum. (b) Corresponding distribution of the global minimizers. (c) Expected improvement corresponding to the prediction. The next evaluation point would be x = 0 (the value of EI at x = 0.8 is slightly lower). (d) Corresponding CEM, which would lead to sample at x = 0.87.

Figure 3 .

 3 Figure 3. Twelve iterations of IAGO on a sine function. (Left) Kriging prediction (bold line) based on three initial evaluation results (squares) and on the twelve additional evaluations chosen by IAGO (circles). The dotted line is the sine function to be optimized (supposedly unknown). The thin horizontal line is the conditional distribution of the global minimum after twelve iterations of IAGO, and its support is almost reduced to the true global minimum. (Right) Conditional distribution of the global minimizers after twelve iterations of IAGO.

Figure 4 .

 4 Figure 4. Optimization of a sine function with IAGO using (9) as the stopping condition, with σ stop = 0.1. Conventions are as in Fig. 3.

Figure 5 .Figure 6 .

 56 Figure 5. Response of an EPSD to a driving test case in the torque -steering wheel angle plane. The dotted line is obtained by simulation for the control laws obtained after optimization of 32 parameters using EGO. The continuous line represents the ideal response according to experts in the field.

Figure 7 .

 7 Figure 7. Mean of the estimation errors ||x *nx * || over 1000 sample paths as a function of the number of evaluations of f , for IAGO (bold line) EGO (dotted line) and a random search (with a uniform distribution, dash-dotted line), when confronted to a irregular Gaussian process indexed on [0, 1] 2 . The standard deviation of the estimation of the mean is always less than 0.02.

Figure 8 .

 8 Figure 8. Minimization of one of the conditional sampled paths used to obtain the results of Fig. 7. The circles indicate the evaluation locations chosen by EGO, the crosses these chosen by IAGO. The order in which the evaluations are chosen is also indicated for IAGO. The global minimizer is at x = (0.77, 0.25) T .


R Léandre
email: remi.leandre@univ-fcomte.fr

The flow theorem for big order generators

Keywords:

Introduction

Dedicated to Professor Volovich for his birthday

Let us consider on R d m vector fields X i without divergence with bounded derivatives at each order which generate the space uniformly. Let us introduce the operator

L = -(-1) k X 2k i (1)
It is an essentially selfadjoint negative operator which generates a contraction semi-group on L 2 (dx) and by ellipticity a semi-group P t on C b (R d ), the space of bounded continuous functions on R d endowed with the uniform topology.

Le us consider R d × G d l , where G d l is the group of invertible matrice on R d . On this enlarged space with generic element (x, g) , we consider

X tot = (X, ∂X ∂x g) (2)
The vector fields are considered here as section of the tangent space of the enlarged space. Let us recall that the Lie algebra of the linear group G d l is constituted by matrices. Let us consider the enlarged vector fields as oneorder differential operators. We can consider the enlarged operator (which is not elliptic in the extra-variables):

L tot = -(-1) k (X tot i ) 2k (3)
It is a negative symmetric operator densely defined on L 2 (dx ⊗ dg) where dg is the Haar measure on the group G d l . It has therefore by abstract theory a positive self-adjoint extension. This generates by abstract theory a contraction semi-group P tot t on L 2 (dx ⊗ dg). If h(x) is a one form on R d , we define h tot (x, g) = h(x).g which is a function from the enlarged space into R d . We can extend diagonally the semi group P tot t such that it takes its values in R d . We have Theorem 1.1. (Blagovenshchenkii -Freidlin-Malliavin) Let us suppose that f and df are smooth bounded

(dP t f ) tot = P tot t (df tot ) (4)
Remark. Let us give the heuristic considerations which lead to this theorem. Over R m , we consider the operator

L f = -(-1) k m I ∂ 2k ∂y 2k i (5) where y = (y 1 , .., y m ) is the generic element of R m .
This operator generates a semi-group

P f t on C b (R m
). Associated to it, there is a formal pathintegral dQ f (w . ) on a "set" of paths w t = (w 1,t , .., w m,t ) starting from on such that if f belongs to C b (R m ) we have "formally"

P f t f (y) = f (w t + y)dQ f (w . ) (6)
We introduce the "stochastic" differential equation issued from x:

dx t (x) = X i (x t (x))dw i,t (7)
such that formally we should get

P t f (x) = f (x t (x))dQ f (w . ) (8)
where f belongs to C b (R d ) This heuristic considerations lead to the the theorem. We refer to [START_REF] Léandre | Symmetry in ordinary and partial differential equations and applications[END_REF] for a rigorous approch.

In the case of diffusion, these considerations are not formal at all. It is the theory of Stratonovitch differential equations in Itô Calculus [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]. But equation [START_REF] Léandre | Bismut's way of the Malliavin Calculus for non-Markovian semi-groups: an introduction[END_REF] has a solution in this case only almost surely defined. Blagoveschchenkii-Freidlin [START_REF] Blagoveshcenkii | Certain properties of diffusion processes depending on a parameter[END_REF] have done pioneering works in the derivation in the starting point of the solution almost surely defined, which were later considerably extended. See the expository paper Meyer [START_REF] Meyer | Flot d'une équation différentielle stochastique In Séminaire de Probabilités XV[END_REF] and the book of Kunita [START_REF] Kunita | Stochastic flows and jump diffusions Probability theory and stochastic modelling[END_REF] for that. We refer to the review papers [START_REF] Léandre | Stochastic analysis for a non-Markovian generator: an introduction[END_REF] and [START_REF] Léandre | Bismut's way of the Malliavin Calculus for non-Markovian semi-groups: an introduction[END_REF] for stochastic analysis for non-markovian semi-groups.

In the first part of this paper, we give the algebraic manipulations which lead to the theorem. In the second part, we solve the problem that we consider a linear test function in the extra-variable which is not bounded by using a Volterra expansion for the semi group in the extra variables. In order to show that the Volterra expansion converges, we use rough estimates of the heat kernel associated to the original operator. 2. Algebraic proof of the theorem Lemma 2.1. Let Y 1 , ..Y l be a collection of vector fields on R d . We associate the collection of vector fields Y tot 1 , .., Y tot l on the enlarged space. Then

(dY 1 ..Y l f ) tot = Y tot 1 ..Y tot l (df ) tot (9)
Proof of the lemma:We proceed by induction on l. We have

(dY f ) tot (x, g) =< dY f (x), g >= < ∇ 2 f (x), Y, g > + < ∇f ∂ ∂x Y, g >= Y tot (df tot )(x, g) (10) because ∇ 2 f is symmetric.Therefore < dY 1 ..Y l f, g >= Y tot 1 (dY 2 ..Y l ) tot (x, g) = Y tot 1 ...Y tot l (df tot )(x, g) (11)
Algebraic proof of the theorem We have

∂ ∂t P t f = LP t f (12) such that ∂ ∂t (dP t )f tot = dL(P t )f tot = L tot (dP t f ) tot (13)
In the expression dP t f we take only derivatives in the initial space. Moreover

∂ ∂t (P tot t (df tot )) = L tot (P tot t (df tot )) (14)
It is the same parabolic equations with the same starting condition df (x).g = df tot (x, g).

Estimates: the Volterra expansion

Let us write

L tot = L + R ( 15
)
where R is a differential operator in x of order strictly smaller than 2k in the original variables and a differential operator which is a polynomial in the A i g where A i constitute a basis in the matrice in R d independent of x.

Basic fact P t has an heat kernel p t (x, y) Let us suppose by induction on n that

P t f (x) = R d p t (x, y)f (y)dy (16) such that |R 1 x p t (x, y)| < C t 1-1/2k 1 t d/2k exp[- |y -x| t 1/k ] (17) if R 1 is a differential
RP t1 RP t2 ..R tn df tot = R t1,..,tn g (19)
where R t1,..,tn is a one form with supremum norm smaller than

C n t 1-1/2k i df ∞ . ( df ∞ denotes the supremum norm of df ).
It is still true at the order n + 1 by the basic fact. In order to show that the Volterra expansion converges, it remains to study if α = -1 + 1/2k

I n (t) = 0<s1<..<sn<t s α 1 (s 2 -s 1 ) α ...(t -s n ) α ds 1 ..ds n = C n t n/2k (20)
where

C n = 0<s1<..<sn<1 s α 1 (s 2 -s 1 ) α ...(t -s n ) α ds 1 ..ds n = C n-1 0<s<1 s n/2k (1 -s) α ds = C n-1 α n (21) But α n ≤ n -α 0<s<1/n s n/2k ds + (1/n) 1+α ≤ Cn -1/2k (22)
Therefore by induction

C n ≤ C n (n!) -1/2k (23)
and the Volterra expansion converges.

Proof of the basic fact

This estimate is more or less classic in analysis (See [START_REF] Davies | Uniformly elliptic operators with measurable coefficients[END_REF]) in the measurable case). We will give a simple proof based on the Greiner rescaling [START_REF] Greiner | An asymptotic expansion for the heat equation[END_REF] and the Davies gauge transform.

We do on R d the change of variable y → x t 1/2k (We operate in x = 0) such the vector t 1/2k X(y) are transformed in X t (x) and tL is transformed in L t which is an operator uniformy elliptic and which don't degenerate when t → 0. By elliptic theory, L t generates a semi-group P t s and P t 1 has an heat-kernel p t 1 (x , y) which behaves well when t → 0. We have

R d p t 1 (0, y)f (y)dy = R d p t (0, y)f ( y t 1/2k )dy (24)
such that p t (0, y) = t -d/2k p t 1 (0, y t

We do for that the Davies gauge transform associated to Ψ i (f )(y) = exp[Cy i ]f (y). We get

Lt = (Ψ i ) -1 L t Ψ i = L t + Lt 1 ( 27
)
where Lt 1 is a differential operator of order strictly smaller than 2k, with bounded derivarive at each order. Lt generates a semi group P t s which has an heat-kernel with bounded derivatives at each order P t s (x, y) and pt 1 (0, y) = p t 1 (0, y) exp[Cy i ] (28) Therefore the result.

RemarkWe correct an error in the estimate of [START_REF] Léandre | The stochastic flow theorem for an operator of order four[END_REF].


Brain-computer interfaces

Origins

Brain-computer interfaces have their origins in Hans Berger's seminal work in electrophysiology on brain waves in 1920. However, it was not until the 1960s that direct communication between a brain and a computer was demonstrated. Edmond Dewan trained volunteers to control certain brainwaves in order to manage a light or send Morse code messages by "thinking" (Dewan 1967). If the power of these brainwaves, measured by an electroencephalograph (EEG), increases or decreases, then the computer understands that the subject wishes to turn the light switch on or off or to generate or not generate a Morse code pulse. A year later, he collaborated with the artist Alvin Lucier, who created a musical performance by making live percussion sounds through his brain activity. At the same time, Joseph Kamiya pioneered neurofeedback, demonstrating that it was possible to regulate one's own brain activity with visual, auditory or tactile stimuli and, in turn, improve cognitive performance. In 1973, Jacques Vidal introduced the term brain computer interface (BCI), which remains to designate all types of direct communication between a machine and a user, passing only through brain activity, the latter being measured and analyzed by the system.

Recent developments in the field of neurophysiological sensors, artificial intelligence and embedded computing have led to a renewed interest in BCI. These neurotechnologies are now expanding in the clinical field (assistive technologies, motor rehabilitation, etc.), in gaming to enhance the gamer's experience and are invading the general public through applications for well-being (meditation experience, sleep improvement). Having become less intrusive, they also offer interesting perspectives for neuroergonomics, through the study of the brain "at work" in the field [START_REF] Dehais | A neuroergonomics approach to mental workload, engagement and human performance[END_REF]. They promote new forms of interaction and promise to improve human performance and its coupling with the machine. Basically, all invasive (i.e. implanted electrodes) and noninvasive brain imaging techniques, such as EEG or even near-infrared spectroscopy, can be used to implement BCI that can be either "active" or "passive" [START_REF] Clerc | Les Interfaces cerveau-ordinateur. 1 : Fondements et méthodes/Les Interfaces cerveau-ordinateur. 2 : Technologie et applications[END_REF].

Active brain-computer interfaces: controling and acting on your environment

Active BCIs allow a user to interact with an artifact (the cursor of a mouse, for example) by generating a "mental command" that will be interpreted by an algorithm. These BCIs were first developed to provide people with motor disabilities with the ability to control effectors, such as a wheelchair, or to communicate by displaying letters on a screen. At present, BCIs do not "decode" intentions or read an individual's mind. Generally, the user must train himself to consciously produce "simple" and "clear" brain signals, for example by relaxing, concentrating or performing mental motor imagery (e.g. imagining a hand movement), in order to induce the right command. In parallel, the machine learns to recognize the said commands, during a calibration phase, from examples of brain signals from the BCI user. Once this human-machine co-learning is achieved, it is possible to interact with a system (a robot, an airplane) to direct it mentally. However, current systems are far from perfect and regularly misunderstand the recognized mental commands. Therefore, they cannot necessarily be used for critical control applications. Another so-called "reactive" approach is to small checkerboards at a certain frequency (e.g. 6 Hz) on a screen. If the subject focuses on the checkerboard at 6 Hz, then their brain response, measured with the EEG, will increase in that specific frequency and the BCI will be able to decode that attention is being paid to it [START_REF] Allison | BCI demographics: How many (and what kinds of) people can use an SSVEP BCI?[END_REF]. Thus, by placing four checkerboards with different frequencies, for example at the top, bottom, right and left of a screen, the user can respectively focus his attention on them to make a robot move forward, backward, turn right or left. The advantage of this technique is that it considerably reduces, or even avoids, the need for a calibration and learning phase. However, the flickering of the stimuli induces eye fatigue in the user and a risk of photosensitive epileptic episodes. In addition, these devices require the user's full visual attention, which is no longer available for anything else, for example, to pay attention to the environment.

Passive brain-computer interfaces: towards "implicit" interaction

In the case of passive BCIs, the goal is no longer to voluntarily control an effector but to use brain activity to enrich the human-machine interaction in an implicit way [START_REF] Zander | Towards passive brain-computer interfaces: applying brain-computer interface technology to human-machine systems in general[END_REF]). The challenge is to infer mental states of an operator, such as his level of mental load, fatigue, stress, in order to dynamically adapt the interaction and maintain an optimal engagement. The idea is then to adapt either the interface (e.g., changing the modality of an alarm to make it more salient), or the sharing of task and authority between the human and the systems, or the operator through different stimulation techniques [START_REF] Dehais | A neuroergonomics approach to mental workload, engagement and human performance[END_REF]. Recent demonstrations of such passive BCIs have been made in real operational conditions to infer the state of human operators such as aircraft pilots. Finally, a last use of passive BCIs, particularly interesting for ergonomics, is the objective evaluation of interfaces and user experience. These BCIs can provide real-time indications to objectively quantify the mental effort, cognitive fatigue or emotional dimensions related to the use of an object or an interface. Many manufacturers in the transportation, video game and social network industries are investing in this approach to ensure increased comfort and safety.

Brain-computer interfaces: challenges

BCI is a promising area of research for many usability applications. However, several challenges remain to be addressed to improve their usability. First of all, BCI requires the wearing of solid electrodes whose discomfort limits their use over a long period of time. The development of new hardware (for example, the use of conductive fabrics) may provide a first step towards a solution, but the design of more ergonomic headset must be considered. Secondly, most BCIs require a calibration phase where the user is required to produce specific commands at specific times so that the algorithm can learn to recognize them. This step can sometimes be long, tedious and must generally be done for each new use. New machine learning algorithms have been developed to increase the accuracy of the classification and reduce the time of the calibration phase. A complementary approach is that of open science (i.e. sharing data on the Internet) to create a large database of neurophysiologic data and train the algorithms on more examples. Beyond this formal work, the improvement of the BCI-user relationship seems to be an avenue too often overlooked by designers. Works [START_REF] Allison | BCI demographics: How many (and what kinds of) people can use an SSVEP BCI?[END_REF][START_REF] Jeunet | Apprentissage utilisateur dans les interfaces cerveau-ordinateur[END_REF] have shown that many individual factors can explain success or failure in mastering a BCI. For example, during the calibration and training phase, the user has little information about their performance (especially when it is poor) and the feedback provided to them is usually low-informative and unimodal (i.e. visual). Training tasks are too often repetitive and directed, and do not adapt to the user. This work recommends building a user model that could predict the user's performance according to his profile (personality, skills, etc.) and use it to customize the design of the BCI and the training of the user to control it. Indeed, controlling a BCI is a skill that should be learned, just like learning to ride a bike. At present, however, researchers do not yet know how to make this learning process effective and useful. Finally, the questions of acceptance of these technologies in a professional setting and the ethical issues associated with them must also lead to a great deal of field research. Thus, ergonomics is a discipline that can not only contribute greatly to the improvement of BCIs, but can also benefit in return from these devices to better understand the relationship between performance, brain and technological environment.