Bedtime storytelling revisited : Le Père Castor and children's audiobooks While audiobooks in England and United States have been popular for a very long time, they are late or even newcomers in France, developping in the last decade of the XXth century with Livraphone (number one publisher with more than 2000 titles), Livrior, Audible and Gallimard being the majors in the field. As in their English counterparts, production consists in classical or contemporary texts being recorded by their authors or by actors, in order to be listened to at home, in cars or public transportation. A quick overview of the public targeted shows that, in France, it was firstly and for a long time intended for children and young people, then developped for languages learning methods for a mixed public and visually impaired people who accounted for 3 millionsso 4,7 % of French population-in 2000. The latest target is now women who represent 75% of paper books readers.

Primary audiobooks were recorded on tapes, then CD's, but the tendency since four or five years is, as elsewhere, to download the book from the publisher's website or online booksellers and to run it on multiple portable devices, Ipods, PDA, Pocket PC or MP3. Despite the wildly popular use of MP3 among high school and college students-(it took only three years for the MP3 in France to equal sales record that the previous walkman had taken ten years to reach), there aren't many educational audiobooks in this format.

Publishers see it as a potential, as suggested by the development of recorded versions of human sciences titles (major classics and today's thinkers), or by a 2005 agreement between Audible and France Loisirs, through which French auditors have access to French audio translations of bestselling thrillers and affiliated such as Sherlock Holmes narratives or Da Vinci Code… Yet, so far in a country that propounds its "cultural exception" and where literature is a landmark, the global number of existing audiobooks hardly matches one tenth -or less?-of the US titles. It is still linked with the tradition of children's narratives: significantly the collection "Ecouter un livre " (Listen to a book) is published by Gallimard, a major literature publisher with a strong children's department; going back to Charles Perrault's Le Petit Chaperon rouge (in Contes de ma mère Loye) first published in 1695, one also finds in the margin of the manuscript an indication of the intended oral voicing of the tale for young people : " On prononce ces mots d'une voix forte pour faire peur à l'enfant comme si le loup l'allait manger. 1 " (These words are to be spoken in a loud voice in order to scare the child as if the woolf was about to eat him).

These basic features about the status of French audiobooks today tell enough how innovative, or even revolutionary, Père Castor audiobook collection was, when it came out in the late 70ies, almost sixty years after the same leading titles of Albums du Père Castor were released on paper. Let us briefly retrace its course of events : Les Albums du Père Castor first came out in 1931, based on the innovative educational movement "Education nouvelle" introduced in France by Paul Faucher after his encounter with the Czech pedagogue Frantisek Bakulé. The first hundred titles were conceived, created, tested and published by a team called Les Ateliers du Père Castor which Faucher had founded. It was soon followed and implemented by Le Centre de recherche biblio-pédagogique de l'Atelier du Père Castor in 1946 and L'Ecole du Père Castor in 1947. After Paul Faucher's death in 1967, his son François Faucher took over, following his father's steps and dedicated approach. It implied sticking to his intellectual legacy and educational values while inventing new forms to enhance it; also, François Faucher had an interest in popular cultures : the pocket book then audiobook versions of the Père Castor's classics were launched, allowing infamous stories -many of them had been translated in several languages, like Michka, with 250.000 copies by 1951 -to reach a third generation of readers in audible versions. François Faucher retired in 1996 and 1 SORIANO Marc, Les contes de Perrault, culture savante et traditions populaires, Gallimard, Collection Tel, 1968, 1977, p. 153. the publishing house, commercially hosted and distributed by Flammarion, is now under the direction of Hélène Wadoswki, a former editor from Nathan, major Publishing house for educational books. New reprints of the classical Albums du Père Castor, in the same original format -a soft cover brochure to ensure easy handling by the child and affordable price to the parents-, are periodically released along with new collections of albums, short stories and novels in paperback (pocket books). As for Père Castor' audiobooks, the move was more daring : other publishers had started producing tapes, but the production was scattered among various producers, some of theme aiming at entertainment rather than literary or educational fiction. No specific editorial concept seemed to sustain it, nor was the equilibrium between narratives and music clearly identified and thought after. So Père Castor's audiobooks was the first recognised and valid book collection to be, as a whole, transferred into audible versions. While pocket books also implied reformatting Père Castor's classics, it mainly led to new authors, thematics and titles, with more text and less pictures.

Audiobooks initially responded to a different approach, technically motivated; square shaped albums, in Italian format, were altered into smaller, rectangular booklets, French format, encased with the matching tape in a plastic cover : it looked like a book, as current DVD's do; the text remained identical, unaltered, but the number of pictures was reduced, their size altered to captions or restricted to only one page, rather than the original spreading of pictures on courir !..." (I am the cookie/pie, the cookie/pie, I was made out of seeds picked from the attic. I was left until I cool but I'd rather run afool." 7 Les Bons amis, by Paul François, told by Yves Barsacq, Marco Perrin, Franca Di Rienzo, Amélie Morin and Mathilde. 8 Petit chat perdu, by Natacha, told by Caroline Rivière and Félix Le Bars. the double-page. Moreover, in several cases, new illustrations were felt needed to adjust to the new product. A new set of illustrations was requested from a different illustrator, so the format and appeal of the booklet in the setting of taped audiobooks likened that of the contemporaneaous pocket book. The "notebook" touch of original albums on thick off white paper that made classics unpretending, simple, familiar thus attractive to children 9 was lost : the booklet inside was smaller, thinner, cheap looking with its glossy white paper.

But as technics evolved, and tapes were outdated, Père Castor audiobooks restored the original album exactly as it was first released, with its very shape, format, paper, illustrations, only slitting a CD (duplicated from the tape or identically re-recorded without any change in style or settings) in the inside part of the back cover. So, even when some of the narrative experience and pleasure of reading was temporarily affected by the cheap format and materiality of the booklet in audiotape series, what was lost on one side got supplemented by the materiality of the sound, voice and music, added. * Thus, transferring albums into audiobooks was not meant as new editorial line open to future unreleased titles nor to titles directly composed and conceived of as audible or "talking books". Rather, it was meant as a process of adaptation and revival of previous titles on paper yet a process wherein the acoustic profile, or audiograph 10 , is strictly consistent with the narrative pattern of the printed version.

The same voice, usually adult and female, tells the whole story and slightly varies in tone and pitch for the different characters.The voice carefully follows the text, it is distinct, clear, rather slow, expressive yet only as the voice 9 One should keep in mind that before P. Faucher's albums, most books and albums for children were Prize books or New Year's books or livres d'étrennes, which were big, heavy, with a hard cover and thick binding that made them hard to open and handle. 10 P. Schweighauser, The Noises of American literature, 1890-1985, University press of Florida, 2006, p. 71: "I define an audiograph as a characterization technique that endows fictional bodies with a set of distinctive acoustic properties designed to position characters with regard to the ensemble of social facts and practices that constitute the fictional world they inhabit." of a narrator, of a distant reader, not of an actor or comedian playing out a part. Unlike Perrault's suggestion, Père Castor's audiobooks are not meant to give auditors a full sense of reality and live, but to convey a vague, metaphorical realism, an imaginary illusion of reality that sets the story at a distance, as the primary book does.

Stories include an encounter between a central character, a child (Marlaguette, Epaminondas, Zohio), a childlike animal (a mouse, a cow), childlike vegetal (lilies), childlike natural object (a water drop), or even a non living object ( toy, pie), and other figures, situations or reality that can be considered adult because it is either big, or unusual, or powerful (bear, wolf, crocodile, wind, sun, mountain, independence, solitude…). Thus, the main narrative voice may eventually be relayed by alternate voices representing various characters, simulating animals or natural objects speaking human language, and delivering their words. Such secondary voices, adult male voices or children's voices, match the various pitches of fictitious voices, yet it is restricted to non threatening, non frightening characters. The storytelling must not impress young listeners and all voices convey a sense of reassurance. So variations depend on the characters depicted and the indications given in the text about their way of speaking or intonation 11 . The character's tone or way of saying may be indicated by the narrator in the narration -"Maintenant, Marlaguette le regardait et sa colère tombait. -Pauvre petit loup! dit-elle. Il est bien blessé! 12 » It can be specified as a stage direction in the clause following a reply : -"Bonne nuit, répondit la Vache de sa belle voix grave 13 "; it can be stylistically suggested by the wording, a naïve set of expression : "Oh !la !la, s'écria le Renard, tu as mangé beaucoup trop 11 For obvious practical reasons, references of following quotations will indicate page number in the printed book, rather than minutes in the audiobook. Although both versions are identical. 12 Marlaguette, p. 11 : "Now, Marlaguette was looking at the wolf and her anger was soon over. -Poor little wolf! She said. He is really injured!" 13 La Vache orange, p. 5 "Good night said the Cow with a deep low voice" d'herbe ! Il faut changer un peu 14 . » Or altogether as in Marlaguette's easy, childish playground talk : "Bien fait! Bien fait! Cria-t-elle en faisant la nique au loup. 15 " Animal's talk " can also be mimicked through onomatopoeia and alliterations and assonances acting as onomatopoeia : -"Si, Si, Si, cria l'Oiseau-Moqueur 16 .." ; -"Hou là! Hou! Cria-t-il en tombant de côté 17 " ; -" Cra !Cra! Cria le geai, il te croquera, Marlaguette 18 ". Or, the animal's cry requires no specific code or alteration from conventional transcriptions: -« La Vache s'assit et répondit : "Meu..meu.., je suis bien malade…" 19 » or « Et la pauvre vache se mit à pleurer. -Meu…eu… Meu…u. 20 » In all cases, dialogues alternate between simulation of animal's talk or cries and illusory human language, both translating the hybrid childlike-animalike speech and thoughts. The audiobook version conveys textual suggestions by selecting an adapted voice for the replies : the capricious Cow in La Vache orange, the loyal wolf in Marlaguette, are voiced out by a masculine voice, while the good helping Fox in La vache orange is voiced out by a child.

Yet, there is no further interpretation, the mere pitch of the voice is enough to distinguish characters. and the audible version does not overcharacterize nor emphasize the narrative pattern nor dialogues : the audible version does not erase the introduction or conclusion of the reply: although character's separate identities are easily deciphered through the intervention of a secondary voice pronouncing talks in direct style, the main narrative voice does not suppress interpolated clauses reporting and describing speech such "s'écria 14 Ibid: "oh!oh! oh! cried the Fox! You have eaten far too much grass! You must change a little ". 15 Marlaguette, p. 11: "Well done! Too bad for you! Marlaguette shouted while mocking, jeering at the the wolf" 16 « If, If, If ', whistled the mocking-Bird, Histoire de Zohio. The bird repeats after the child hesitations between different routes or actions "If I take this way, I'll meet the buffalo, If I take that way…" 17 A wolf knocks his head against a rock at the entrance of his den while holding Marlaguette by the skirt in between his teeth. But the comic, wailing sound "hou là! Hou" suggesting he hurt himself, as in the interjection "ouille/ouch" is a slight and mild distortion of the conventional longish, frightening howling of the wolf "houuuu…" (Marlaguette, p. 10) 18 Ibid., p. 19. The "croaking" jay warns Marlaguette that her friend the wolf will ultimately « crunch » her. 19 The cow sat and answered : Moo.. moo.. I feel really sick.. La Vache orange, p. 4. 20 Ibid. p. 20: "And the poor cow started crying: -Moo.. oo.. Moo.. oo." le Renard; répondit la Vache 21 ". The narrator's voice is a reader's voice that doesn't derail from the original score, just as one would do in reading aloud a story to a child after the printed book. So in the audiobook voices recorded and the general narrative tone do not assume extraliterary significance.

Throughout each story, a short musical line, generally a simple combination of a few instruments at a time among which flute, xylophone, guitar, trumpet, drums, clavichord, -, is being identically repeated. It instantly punctuates the storytelling, as when an oral storyteller takes a breath, and replaces the setting of the page: music discreetly marks blank transitions between actions, indicate the child when to turn the page, and give him the time to do so. During narrative phases, melodies either stop or accompanies the voice as a musical backing, still barely audible in the background. Although tunes are simple, and rhythm binary as in circus music or brass band march, it is not easily graspable as a refrain or chorus, so it works as a jingle not as a musical speech per se. Melodies or jingles vary from one audiobook to another in Père Castor's Classiques, equally easing the storytelling and listening process like a beatbox.

Finally, there are a few "noises" in Père Castor's audible versions.

Obviously, as in any careful and professionnal recordings, noises are intentionally part of the sound track. They do comply with Schweighauser's introduction to a communicational approach of noise that instead of conceptualizing it as a disturbance impending, masking communication, presents it as a signal that increases information 22 . Indeed, noises in Père wonder what the purpose of such sounds can be, as they mimick or echo the character's actions?

In La Vache orange for example, the central scene between the big sick orange cow and the little grey fox who nurses him is rhythmically punctuated by such sounds : Plastic chewing when the cow sucks a baby bottle to check his fever; huffing and puffing to suggest cow's sickness; running water when the fox fills the hot water bottle to soothe the cow's stomach; "atishoo" when the cow sneezes after drinking champagne; rumbling and whistling for the train in the cow's dream; birds singing and cookelandoo next morning; crunch crunch for the toasted bread etc. Sounds follow and duplicate the actions narrated, yet one cannot speak of redundancy. As sound effects use a different language or communicative system, acoustic versus verbal, they appear as part of a special language brought up by the audible version to implement the original story. Rather than repeating narrated replies and facts, they function as resonances introducing another level of communication as in live narration. The effect is playful : Indeed, indexical recognizable sounds do not introduce a conflict or counterpoint to the narrative as they are neither simultaneous nor as loud as the voice. They echo the narrative on a low key and seem to confirm the tale with an undisputable or at least fair proof, albeit comical. Studies on reader's response to Père Castor's audiobooks, or to more recent, popular yet low-brow Marlène Jobert 23 's audiobooks show that children insist on a strict respect of the written sentences, specially its punctuation. A proper narrative tone, in the young listener's idea, is one that follows the ups and downs of the sentence, marks the dots, commas, exclamation points or question marks. The child's pleasure, in listening to an audiobook such as Père Castor's, where a genuine book is attached to the CD, relies on respect for the written narrative and strict conveyance of its intended meaning. It is also a combination of watching, reading words and pictures and listening. It relies on duplicating the written story by the narrative voice.

As opposed to noises in the "simultaneous poem" described by Douglas Kahn 24 and which "represent the background-the inarticulate, the disastrous, the decisive, [showing] in a typically compressed way […] the conflict of the vox humana with a world that threatens, ensnares, and destroys it", noises in Père Castor's audiobooks are reassuring means of a reconstruction: superfluous, they slow the narrative process, assess its verbal statements as gestures acting out speech, thus empowering the three dimensional audiobook, giving weight and thickness to the third, acoustic dimension of the album.

Yet, Père Castor's audiobooks take only limited advantage of the possibilities presented by phonography and acoustic tools. Obviously and deliberatly, rather than for a lack of know-how or for economical reasons, they resort to sound perceptions and sound effects to emphasize narration and to the extent that it doesn't interfere with the storytelling. In this line of thinking, sound effects are utterly intended and distinct on the blank band both from the voice they accompagny and from musical lines; as such their effect is to set the listener within a specific frame of mediation. Through artificial, almost theatrical sound effects, the process of storytelling is underlined and catches the attention of the listener. Cries and noises are obvious imitations, unrealistic mimicks that stage the representation process itself. In freely adding unwritten yet non verbal lines, the audiobook emphasizes the communication process and the acoustic tools on which it relies. While its goal is theatrical and may induce listeners to laugh or add credibility to the scene, it also creates an attractive acoustic presence that seduces the child. As a real adult reading to a child draws and keeps the child's attention by the power of one's actual, real, physical body and voice, sound effects, noises act along with alternate voices and music to sensually embed the narrative. Castor made it possible, worthwhile, logical -or even necessary-to turn these 1931's to 1950's albums into audiobooks? Secondly, which "immediate needs" of the child do Père Castor's classics audible versions answer? Père Castor's classics whether as a single book on paper or as book plus CD, are "albums", a word designating books where pictures equally share the space of pages with the written texts, or take over words as defined by the English "picture book". The word "album" has several meanings, (from the latin "albus", white space for inscriptions and listings) and is used in French for 25 ans", and Jules Hetzel's 1866 "Albums Stahl". But he is considered the founding father or pioneer of the type in France : he conducted a systematic study of the album and invented a radically innovative approach to its format, text and pictures interactions that refers to a deep understanding of the child's personal development along with the evolution of his reading practices. The album is seen as an "iconotext" 27 wherein meaningful effects rely on the interaction between text, pictures and their material support of transmission, format, paper, texture, weight, handiness. Père Castor's classics are narratives, genuine stories, not fairy tales or legends originating from an oral tradition. They respond to a natural perception of events as a drama script, a theater play. Yet, they are also written to be spoken or sung or to be read aloud by parents to children. The sound track adds a new material aspect, yet as the child keeps looking at the album while listening to its story, the iconotext is not undermined but supplemented by the voice.

Paul Faucher acknowledged this given from the start, publishing his Classiques under the pseudonym of "Père Castor", literally "Father Beaver", 26 where the beaver, as in "Father Christmas" is a reading beaver or (as in later versions, DVD adaptations of the albums) an anthropomorphic spectacled figure, half animal, half grand father. As suggested by the "Father Beaver" brand name, pseudonym and logo, where the beaver alternatly stands as the young reader/listener or the adult reading to him, les Classiques du Père Castor belong to a moral and educational approach favouring a positive, constructive approach to one's life and the transmission to younger generation of a deeply rooted, unwritten wisdom based on the observation of nature, animals and countryside lifestyle. Père Castor's primary albums for children's activities, then tales, then documentaries on animals, were all meant to answer to the child 's narrative intelligence, as the american psychologist Jerome S.

Bruner would later state it, that enables him to decipher other people's intentions and state of mind, to put oneself at a distance, look at the world from another point of view and understand someone else behaviour as that of an alter ego, both similar and different. Before four years old, the child is not able to decenter himself in order to attribute people beliefs that would be different from his own. Yet developping this skill allows him to build himself and the others as characters. Thus narratives are a fundamental way through which we give meaning to both our life and that of others 28 . 28 Jean Molino and Raphaël Lafhail-Molino, Le récit, un mécanisme universel, in Contes et récits : pourquoi aimons-nous les histoires?, Mensuel N° 148 -Avril 2004.

In Paul and François Faucher's line of thinking, the Album is an open, not preformatted object, meant for multiple purposes, speak, play, discover, learn, imagine. It addresses young children with little or even no reading ability. It must be brief according to the child's concentration capacity.

Stories must be simple, easily understandable, with pictures sustaining the understanding process. Both texts and pictures are thoughtfully conceived and produced: Paul Faucher carefully chooses writers and artists from the avant--garde and from Eastern Europe graphic school (Nathalie Parrain, Gerda Müller) to write and illustrate the stories. It takes about a year to publish a book less than 20 pages and the adjusting process is essentially collective : the project is discussed in team including the illustrators and writers, the graphic, publishing, and printing production group, along with pedagogues and children --young readers or pre--readers--whose reactions to the various drafts influence choices and decisions.

A similar process applies to audiobook versions. Consistently, as recurring authors and illustrators often sign several titles on paper (Paul François, Natacha, Gerda Müller..), the same voices or storytellers record several audio titles (Catherine Le Bars, Franca di Rienzo for female voices, Yves Barsacq, Marco Perrin for male voices); children, only mentioned by their first name, are also often associated to narrate or play parts in the recorded versions of several stories (Sebastien, Mathilde, Chantal, Élodie, Fred, Julien, Kathy, Loïc, Marion, Natacha, Régine, Sylvia, François 29 …) Pocket books and audiobooks thus result from a collective process and gathering of "book people", educators, psychologists, all sharing the same interest in the child's present and future being.

Secondly, Père Castor's albums respond to an educational approach: the objective, as stated in an address to the adult mediator at the beginning of some specific albums, strictly "picture books" without text, is to induce the child in spontaneously and orally wording his reactions to the story. With the help of an adult, the child will "have the pleasure to recognize, and name objects he knows; identify and attach specifics to characters; analyse and explain, picture by picture, actions, movements and gestures; interpret expressions, looks; compare pictures and establish by deduction the chain of events. Once he has appropriated the various elements of a story, he will build it, add to it as he wishes and will tell it in his own way with the ease he will have acquired during your common "walks" through these pictures 30 . "

An analogous educational objective is assigned to the "illustrated" book, meant as "Secondes lectures 31 ", following the child's physical and mental developpment. When the child masters the three first phases of picture reading, enumerating or naming, describing and interpreting, then the fourth where he differentiates and compares, he is ready to learn how to read. Thus, Paul 29 In the audioversion of Anne-Marie Chapouton, Les lettres de Biscotte Mulotte, Marie Colmont, Marlaguette, and other titles. Faucher states, while a more elaborated and focused type of reading appears, illustration becomes a substitute for pictures, it conveys the general context, describes the atmosphere, announces an action to come, completes the written narrative, but the main part of the story is narrated by the text 32 . Whatever the age of the child considered, the album, whether it is a picture book or an illustrated book is the means to answer children's needs, to connect children because its conception relies on mixed considerations, psychology, applied pedagogy, literature, graphic art and editorial techniques.

In this line of thinking, Père Castor's audiobooks, albeit with additional techniques, must answer similar needs and concerns. Père Castor's albums on paper were meant are the basis of a verbal and oral narration, a verbalisation of pictures initiated by the adult, possibly imitated and rehearsed with him, then ultimately carried out by the child on his own: while induced to read aloud to himself a book open in front of him, the child reader stands as both the listener and independent producer, "voicer" of a home made, free style and unrecorded audiobook.

Books altogether are the means of mental representations transmitted by an author to a reader through a set of signs. As shown by Reader's response theory, a text gains meaning by the purposeful act of a reader reading and interpreting it. The relationship between reader and text is highly valued. It isn't just that texts do not exist without readers; it also implies that reading is not a one-way, bottom-up process, from the book and implied author to the reader; it is also a top-down process, as the reader himself produces mental representations of his own, based on his experiences or personal context, thus imposing or superimposing his response to the author's message: reading can be viewed as an individual reconstruction process within the constraint of a written text, a process that affects the text. This is specially true for children's reading, not because we may see it as an interpretative community, but merely because of the child's cognitive way of reading : a physical, practical, functional, unconventional gesture of handling, relating to a book, that implies deciphering, decoding words and pictures, but also many other operations like touching, laughing, jumping, imagining, extrapolating, identifying to the characters, playing a part in the narrative … Reading for a child is the means of self-construction, of building one's identity through a variety of methods among which consciously focusing to understand a specific story, an author's message or a book's meaning is a secondary issue.

On the other hand, repetition, re-reading is a typical pattern and method of a satisfactory self-construction process which Père Castor's audiobooks respond to.

As Italo Calvino states in Pourquoi lire les classiques 33 , "classics are those books about which one always says : I am re-reading it.. and never: I am reading it, those that constitute a valuable for those who have read and loved them as much as for those who postpone the pleasure to discover them fir the first time in better conditions; a classic is a book that has never finished saying what it has to say 34 ". Père Castor classics are classics according to Calvino's definition because they shape children's future experiences, providing examples, role-models, terms of comparison 35 and many other elements which remain active and influential for a long time, even when adults have forgotten a book read during childhood. Also because they are never just read for once and for good but always read, told, narrated, leafed, thumbed, flicked, glanced through again and again and transmitted from one generation to the other.

Reading a favorite story for a child is repetitive process of re-reading it or, before reading skills are mastered, a process of re-hearing it again and again 33 Italo Calvino, Perchè leggere i classici, Mondadori, 1991, french translation Pourquoi lire les classiques?, Ed. du Seuil, 1984, 1995, p. 7-14. 34 Our translation. Ibid., p. 8-9. 35 Ibid., p. 8. from a familiar voice which Père Castor's audiobooks indulge. The most obvious proof of it being the fact that at L'heure joyeuse, copies of Père Castor audiobooks are out of use and have to be renewed after being borrowed by only ten readers at the most: each reader or hearer must have listened to it extensively. Just as one likes playing again a favorite tune, young audiobook listeners like to play again a favorite story. Each hearing adds to the previous in order to build not just the child's general literacy but rather an "imaginary text 36 " that has become an intrinsic part of the child living and educational experience: it consists in the basic story itself progressively supplemented, superseded, enriched and transformed by the diversity of private or personal hearing sessions to which it has been submitted.

*

Altogether audiobooks comply with the interest in developping the child's linguistic skills and his need for security, both emotional or affective and factual or concrete ; they are also meant to enhance his ability to think freely and by himself, to express tastes and judgment of his own. Thus, they also comply with a communicational approach of books for children, that is to say an approach that is not only based on a practical subject to object relationship, but on a pragmatic subject to subject relationship 37 . Moreover, the subject to subject, pragmatic relationship offered by the book is duplicated in the case of child's reading, by the adult's intervention or mediation : to non reading infants, books are identified as objects voiced out by their parents or surrounding adults; they first learn how to read by imitation, hold the book upright, even if upside down, follow black lines with their fingers 36 A term proposed by Deborah Vlock quoted in M. Rubery's "Play it again Sam Weller", op. cit., p. 61. 37 As defined by Daniel Bougnoux in Introduction aux Sciences de la communication, La Découverte, Coll. Repères « Culture-communication », 1998, 2001, p. 9. and produce accompanying sounds, from burble to phrases 38 . More than that, they attach the book to the voice that propells it, sometimes touching the adult's throat as an intrinsic part of the device, and were it not for pictures, the album would appear as a record which requires a human player to be listened to.

Oddly enough, as Daniel Bougnoux 39 notices, while the longlasting era of printed books, or "graphosphere 40 " is being replaced with the new order of videosphere, many fear the separation of culture from print and print culture.

Yet, the printed book, inky-black on white, with its regular, justified lines of type-settings and Block letters, is among the most severe, stern, ways of representation, one that retains little of the sensory aspects of oral speech. The reader is isolated and as less is more 41 the sensorial soustraction and typographic austerity of the book is the means of a mental, if not spiritual liberation: his consciousness focuses on the inner message of the book, enticing him to developp an individual analysis, a critical approach of knowledge, an ability to imagine, extract and recompose a world from the black and white symbols.

Obviously, children' books tend to alleviate this sense of isolation and lack of outside seduction: attractive, colourful covers and pictures break the linear, rigid succession of words to be read and allow a free, random, multidirectional circulation of the eye on the visible space of the page. Texts and pictures congregate and combine each other in children's picture and illustrated books to sustain the child mental representations. Pictures and illustrations also compensate for the relative dryness of the printed text and as the impact of images is stronger, its potential danger or affect greater for a young sensitive mind, they also tend to represent situations and characters in a milder or metaphoric way. While the third meaning thus created may be specific to each individual reader, the general purpose is to convey a sense of security, to relate 38 L'enfant lecteur, Rolande Causse (dir.), Ed. Autrement, 1988, p. 71. 39 « L'explosion communicationnelle », Ibid., p. 92-93. 40 As defined by Régis Debray in Cours de médiologie générale, Gallimard, coll. Textes essentiels, Paris, 1991. 41 As Nicholas Negroponte, after Mies Van der Rohe, used to say. the child to a circumscribed world wherein one feels at home and where reality only enters drop by drop.

Children audiobooks operates accordingly. While one could take for granted that a decrease in reading abilities, time or taste rates one among specific conditions attached to the success of contemporary audiobooks, children's reading studied through the consideration of its real or imaginary environment and goal, bring forward unexpected foundings about how and when young readers resort to audiobooks rather than to print, or to both audiobooks and print.

First, children's audiobooks, as other new technology of transmission operate as a filter: they create an intermediary reality, neither inside nor outside 42 . Second, they relay and supplement traditional print works: the voice is an indicator of the transmittor's body that pilots the message and counteracts abstraction ; the illusion of a presence, of direct, albeit defered oral speech, adds to the seductive, soothing effect of the pictures on the attached booklet.

Third, audiobooks exercise the auditory imagination 43 because they restore or replicate a communicational and relational experience that is intrinsic to the child's primary narrative experience or to an acquired experience wherein knowledge, culture, imagination are linked, relayed and transmitted by way of the book. One could object that, as a matter of fact, while this must have been the primary motive for the audiobook series, social studies tend to show that the practice of reading aloud a book to children or vocally narrate a story is not so much in use as before; one of the reasons involved being children's early access to a diversity of listening, watching, interactive playful devices. Yet, technics and technical objects remain social connections, albeit as surrogate, and social 42 Daniel Bougnoux, Introduction aux Sciences de la communication, op. cit., p. 70. 43 The expression is borrowed from a presentation by Dr Joy Alexander, Queen's University, Belfast, "Audio relationships are shaped and conducted by technical devices [START_REF] Bougnoux | Contenir le réel[END_REF] : indeed, as we have seen with examples from Père Castor's Classiques albums in audible versions, audiobooks enhance the relational function and purpose of the album rather than diminish it, despite the lack of a real body voicing out the story. As suggested by Matthew Rubery [START_REF] Rubery | Play it again, Sam Weller[END_REF] , hearing a book through an audiobook has to do with the "Perpetual immediacy" recommanded by actor Barbara Rosenblat "to recreate as far as possible the childhood experience of hearing stories read aloud [START_REF] Rubery | [END_REF] " and it "has been described in synaesthetic terms through an aphorism formulated by Murray Schafer; "hearing is a way of touching at a distance" 47 ".

Fourth, resorting to audiobooks is not the way in which "our age translates itself back into the oral and auditory modes because of the electronic pressure of simultaneity [START_REF] Mcluhan | The Gutenberg Galaxy[END_REF] ." On the countrary, it is precisely the electronic pressure of simultaneity that sustains the resort to audiobooks for children: reading a printed book is an activity of its own that excludes doing something (except listening to music in the background); listening to an audiobook can be done along many other tasks or endeavours, from travelling, walking, drawing, cooking, playing to… reading another book or printed material! Although this takes us afar from Père Castor Classiques, which are short, "minutes" album (from 5 to 8 minutes at the most) for younger children dedicated to story time and intense listening, a study led by a leading french documentation center and children's literature library [START_REF]La Joie par les livres[END_REF] with a consistent children's audiobook section, reveals that many heavy audiobook consumers tend to read a print book or rather a comics while simultaneously listening to an unrelated story on audiobook.

Fifth, as Joy Alexander 50 finds in her own survey of pupil's response to audiobooks, 39% of 120 1st and 2nd years, 12% of 65 3rd and 4th years, 22% of 125 sixth-formers said they currently listened to audiobooks, while respectively 38% and 48% of the two latter groups said they had been in the past and might prefer print. Asked about the merits and limits of specific titles (H. G. Wells'

The War of the worlds, Steinberg's Spoon face, and what they liked or didn't like about the experience, their answers show a strong interest in the acoustic medium -" it is like other people reading for me.", I liked the way it sounded so realistic.", "I liked it because it sounded so believable.", "I enjoyed it because it seemed very real and even though it was on the radio you could picture nearly everything.", "I liked it very much. I thought it was very life-like and very interesting to listen to it.", "It was fun to listen to." Yet they point to an equally and reverse preference for reading print because of negative aspects of the audible version: "I don't really enjoy it because they don't sound the way I think they do.", "I like to read in my mind and create my own voices for characters.", "I would prefer reading a book at your own speed.", "I listened to them in primary school and I found them boring. It is much better to read it yourself."While some answers illustrate hearing habits and practices of new generations -"You can just concentrate on listening."-, or current sense that reading print requires an effort anybody is not likely to provide -"I don't like reading -only if it's a good book. Listening was better.", it is intriguing to find suggestion that something might be lost in the audiobook experience 51 ; that what they call "listening" might in fact be solely "hearing", and that it may impair children's imaginary skills : "I feel listening to a story you need a good imagination. I probably would like reading better." So altogether audiobooks still exercize the auditory imagination by inducing the transient state of mind and distant approach to reality that was brought about before by the sole printed books. * These findings finally point to a distinctive purpose of reading and specially child's reading which both audiobooks and print books tend to : reading or hearing books as play.

For children reading is not just a leisure but a game. It may occupy from a small to the main part of their playing time. This is also true for many adults for whom it may replace playful activities such as playing cards, crosswords, handcrafts… It is a game with literary effect as suggested by several critics 52 , yet no one had really attempted a scientific approach of this game until Michel Picard's book La lecture comme jeu 53 . Picard observes that reading for children is often associated with writing and drawing, but his analysis goes beyond the case of children's reading and resorts to various theories to present reading as a game obeying a set of rules and to outline the issue and effects of reading experiences considered as playing, playful experiences. Although it would decidely be quite an interesting track to follow and apply to specific cases of 51 As one of the essay's reviewer pointfully suggests "some important things about reading might be lost in the audiobook experience, whatever the gains in vividness and presence. One of the toughest issues is the question of the reader's passivity in relation to recorded as opposed to printed texts. One contributor, David E. Beard, even intimates that "the process of listening to an audio book (rather than reading a printed book) actually makes us less able to grapple with the complexity of ideas possible in the text; we enter into those newly possible social relationships at a deficit." audiobooks and children's audiobooks in particular, it is an untimely purpose for this paper to enter at this stage. Let us simply lay out its first, basic guiding principles after Johan Huizinga's infamous essay on play 54 . Trying to understand play as a cultural factor in life, Huizinga observes that "the great archetypal activities of human society are all permeated with play from the start 55 " (such as language which is playing with this wondrous nominative faculty, playing upon words); that "although it is a non materialistic activity it has no moral function 56 " yet cannot altogether be included in the realms of the aesthetic, despite many and close links to beauty: So Huizinga leaves it at that "play is a function of the living, but is not susceptible of exact definition either logically, biologically or aesthetically 57 ".

Significantly enough for our understanding of book reading/hearing as play, Huizinga explains his tactful approach to the notion of play by stating that one should "take play as the player himself takes it: in its primary significance 58 ", and developps his statement with an example close to book and storytelling experiences: "If we find that play is based on the manipulation of certain images, on a certain "imagination" of reality (i.e. its conversion into images), then our main concern will be to grasp the value and significance of theses images and their "imagination". Yet, while the latter example equally relates to both print and audiobook, it seems more accurate to apply to book hearing -rather than its elder book reading -and to children's audiobooks, Huizinga's definition and approach of play: -a free act, felt as "fictitious" and remote from daily life, yet able to fully absorb the player, orderly acccomplished 54 Picard himself recalls Huizinga as a starting point of his essay : Johan Huizinga, Homo Ludens -Essai sur la fonction sociale du jeu (1938) in a circumscribed time and space, and obeying fixed rules, and inducing group relationships often surrounded with mystery. Along Huizinga's line of reasoning, children's audiobooks as middle or low brow cultural objects (despite highbrow quality of texts and literature might convey) belong to the game-play category. They consist in a voluntary activity, done at leisure, during free time. Whether they are or not considered educational by parents and educators who provide them as part of a nurturing programm to developp necessary faculties, audiobooks are not bound up with notions of obligations or duty. They imply "stepping out "real" life into a temporary sphere of activity with a disposition of its own 59 ": as every child, says Huizinga, knows perfectly well that he is "only" pretending, the conjuring effect of technics emitting, receiving a recorded voice sustains the pretense. Yet, as for play, "that doesn't prevent it from proceeding with the utmost seriousness with an absorption, a devotion, that passes into rapture and, temporarily at least, completely abolishes that troublesome "only" feeling. 60 " Most of all they are enjoyable: "The need for it is only urgent to the extent that the enjoyment of it makes it a need 61 ."

What then is the value of playing Père Castor's audiobooks ? They remain special editions, as they are restricted to the Classiques Albums series. With time and social changes, they have become literary and also historical landmarks. In today's urban societies they relate to a rural life that children no longer know about. Seemingly oldfashioned they remain plausible; they exercize the reader's ability to imagine ; they still provide a reassuring visual, narrative, acoustic context; eventually they convey a message of hope and a sense of responsibility for the child' s life and endeavour. The maternal, somehow naïve 59 Ibid. 60 Ibid. 61 Ibid. point of view they adopt strongly contrasts with present children's series wherein young readers are confronted with the same harsh contemporaneaous issues themselves and adults are facing in real life. The idea that books should allow and help children to grow autonomous in a protected environment, adapted to their scale and size and needs is a distinctive feature of Père Castor's and one that distinguishes his editorial approach from other major contributors to quality literature for children; even from François Ruy-Vidal and Harlin Quist who appeared as distant and original followers of Père Castor's line of thinking in the 70ies and 80ies. Yet, it is remarkable to observe that despite or possibly because of the sense of affective security and the surraneous setting of fictions which contradict present narrative and living experiences of XXIst century children, Père Castor's fiction still offer a window to the world. How little its likeness to the current world heavily portrayed and documented in today's medias the world in Père Castor's audiobooks remains valid : it is an accurate understanding of children's lasting exposure, one generation after the other, to an adult surrounding, an unknown, sometimes disturbing or harsh reality; and retains a sound philosophical value in strengthening the child while suggesting both resistance and enthusiasm.

Père Castor audiobooks have inherited from educational theories which they now broadcast. They relay a conventional and affective ritual of bedtime storytelling. While the ritual is revisited by technical tools, it is not completely altered nor disembodied. An obvious proof being that the series's leading title Michka displays the founding father -Père Castor-Paul Faucher's very own voice 62 ! On the contrary, while pictures in print albums have aged, voices in audiobooks update and regenerate stories and a cultural phenomenon of 62 Probably recorded as part of the tests conducted in l'Atelier du Père Castor, before his death in 1967.

  The audible versions of nineteen titles 2 from Les Classiques du Père Castor all originate from the collection conceived by Paul Faucher and the Père Castor collective workshop. Recorded on tapes upon François Faucher's initiative, they were then progressively and even recently released on CDs. They are currently marketed by Livraphone and other online specialized bookshops such as "Le livre qui parle" (The book that talks); periodically out of print and reprinted (paperbooks and CDs as well), several are obviously continuous bestsellers: La Sieste des mamans 3 (Mumies'nap), Epaminondas, Les trois petits cochons (Three little pigs), La grande Panthère noire (The big black panther).The most famous of them, pledged by one generation after the other, are Marlaguette, LaVache orange (The orange cow)4 , Michka 5 and Roule Galette (Roll pie/cookie)6 . Some are adapted in english versions as language learning tools, such as Les Bons amis translated as Good friends 7 , and Petit chat perdu translated as Little lost kitten 8 . Publishing Les Classiques du Père Castor and new Père Castor titles in pocketbook was a major step and change in the editorial policy. It reflected a perception of new economic and reading practices which several children's literature publishers had in common and that is still accurate today.

  Castor's audiobooks are sound effects added to the audible version, knock knock or rat-tat-tat to suggest footsteps, rustling noise of crumpling fabric to suggest a fall… The narrative being clear and self-sufficient, sound effects are not necessary to complete it, nor to supplement its meaning. Then one can 21 "The fox shouted; The cow answered."22 Refering to Claude E. Shannon, and Shannon and Weaver, in The Noises of american literature, op. cit., introduction, pp. 6-8.

  Let's assume that Père Castor's audiobooks, as other children's audiobooks, induce young listeners attention and contribute to their pleasure, if not by creating or recreating bedtime storytelling, at least by creating an aural, vocal setting reflecting previous experiences of stories, picture or illustrated books being passed on to the child through the mediation of an adult voice. * True to the inheritage of his father (P. Faucher) and godfather (F. Bakulé), François Faucher considered how to adapt to changing times, after 68 and the start of modern era in children's literature. In 1975 he declared : "Sur le fond, je reste fidèle à la pensée de Bakulé et de mon père, pensée fondée sur l'observation, le sens de l'humain, l'éducation de l'imagination, du caractère et du jugement, l'invitation à la création, plutôt que sur l'acquisition de connaissances par trop éloignées des besoins immédiats 25 ." So what? which specific, intrinsic aspects of Les classiques du Père

  Quoted by Marc Soriano in Guide de la littérature de jeunesse, Paris, Flammarion 1975, p. 241. « I remain true to the theory of Bakulé and my father, based on observation, sense of humankind, education of the imagination, of the attitude and judgment, on the incentive to create rather than the acquisition of notions way too far from immediate needs." multiple objects such as photography book, music book, songs book or record… In albums at the turn of XXth century, as in 1883 (Vieilles chansons et rondes pour les petits enfants) et 1884 (Chansons de France pour les petits Français), by the painter Maurice Boutet de Monvel (1851-1913), published by Plon, the graphism is linear, simplified, without perspective, colours are soft or pastel and flat as in japonese etchings ?, to serve a clear, easily readable picture, to promote good taste and an idealized vision of childhoodness 26 . Text is secondary while pictures and visual typography convey the essential part of the narrative. Paul Faucher may have inherited the notion of album from XIXth century attempts such as Louis Hachette 1861's "Albums Trim pour les enfants de trois à six

  Annie Renonciat, BnF -Babar, Harry Potter et Cie. Livres d'enfants d'hier et d'aujourd'hui. Expositions.bnf.fr. 27 I. Nières-Chevrel, Introduction à la littérature de jeunesse, Didier Jeunesse, coll. Passeurs d'Histoire, 2009, p.118. « Il désigne l'ensemble des livres pour enfants dans lesquels l'image prime sur le texte, mais également -à l'intérieur de cette désignation éditoriale globale -ces livres dont les effets de sens reposent sur des interactions du texte, de l'image et du support, et ce qui sont ce que l'on appelle des « iconotextes. »

  30 « Il prendra plaisir à : reconnaître et à nommer les objets qu'il connaît ; identifier et caractériser les personnages […] ; analyser et expliquer, image par image, les actions, mouvements et gestes […] ; interpréter les expressions, les regards […] ; comparer les quatre images, rétablir, par déduction, les actions intermédiaires et l'enchaînement des faits. […] Quand il aura pris possession de tous les éléments d'une « histoire», il la reconstruira, l'enrichira, à son gré et la racontera à sa manière, avec l'aisance d'élocution qu'il aura acquise au cours de vos « promenades » à travers ces images. »

  books and young people: exercising the auditory imagination", The Irish Society for the Study of Children's Literature, 2010 Conference: 'Sound Image Text' , 5 th & 6 th March 2010, Arts Block, Trinity College, Dublin 2.

  retracing the history of L'Education Nouvelle (1899-1939), movement which inspired Paul Faucher's Père Castor publishing work. 50 Dr Joy Alexander, Queen's University, Belfast, "Audio books and young people: exercising the auditory imagination", The Irish Society for the Study of Children's Literature, 2010 Conference: 'Sound Image Text' , 5 th & 6 th March 2010, Arts Block, Trinity College, Dublin 2. The survey was conducted in 2009.

  storyhearing or voicing over the child'shoulder which answers the needs and Habits of the Heart 63 . Brigitte Ouvry-Vial 63 Borrowed on purpose from the title Habits of the Heart: Individualism and Commitment in American Life, Robert N. Bellah, Richard Madsen, William M. Sullivan, Ann Swidler, Steven M. Tipton, U. of California Press, 1985.

  Plaisir du texte, Seuil, 1973, rééd. Points 1982, p. 11 ; Holland, Norman, The Dynamics of Literary response, NY, Norton, Oxford University Press, 1968 ; « Game, play, literature », Yale French Studies, special issue, Yale University, New Haven Connecticut, 1968… 53 M. Picard, La Lecture comme jeu, coll. « Critique », Minuit, 1986.

52 R. Barthes,

  , Gallimard, « Les Essais » 1951, réed. 1976, trad. C. Sérésia, p.35. We will refer to the english version : Homo ludens, A study of the play element in culture, 1938, Routledge and Kegan, 1949, Routledge, London, 2002.

55

Ibid., p. 4.

56

Ibid., p. 6.

57

Ibid., p. 7.

58

Ibid.

Full list of titles available on livraphone online bookshop.

La Sieste des mamans, by Bertron : exhausted by their youngsters, their meals, naps, and cries, mother crocodile, mother elephant and mother monkey protest and claim for themselves the right to have a nap. When they wake up, worried about what may have happened, it turns out the kids have dealt and taken care of themselves.

The orange cow, a story by Nathan Hale, told by Catherine Le Bars et Olivier Foy: the gentle and kind fox brings to his home an orange cow that feels really sick. But is not easy to take care of a capricious cow who won't take her medicine..

Michka, told by Paul Faucher et Sébastien, the teddy bear, leaves on his own into the woods. What a bliss to be free! He hears geese mentioning it is christmas night and everyone should do something good. So when he arrives with the christmas XXX at a sick and poor little boy"s house, and there are no toys left, Michka sighs deeply, enters and sit in the empty shoe, awaiting for the child to wake up.

Roule Galette, by Natha Caputo, told by C. Le Bars et S. Benmerzoug. The word "galette" refers to a pie-size thick cookie or bread cake, and also to traditional oatmeal savoury pancakes from Brittany. The formula" Roule Galette" has even become a trade mark as it recently inspired the name of a new fastfood type chain of restaurants serving savoury pancakes... astory of a cookie itself free, encountering several predators (bear, wolf..) and escaping them until the fox is wiser. At each encounter it shows of with a little song or rhymes :"Je suis la galette, Je suis faite avec le blé Ramassé dans le grenier. On m'a mise à refroidir, Mais j'ai mieux aimé

A former movie actress, Marlène Jobert has recorded numerous stories for Hachette audiobook series. Quite popular among children, her recordings are less favored by parents who find the style of her narratives closer to popular and low brow entertainment than to quality literature.


Nikolai Vatanski
email: nikolai.vatanski@hut.fi

Jean-Philippe Georges

Christophe Aubrun

Eric Rondeau

Sirkka-Liisa Jämsä

S.-L Jämsä-Jounela

Compensating the transmission delay in networked control systems

Keywords: Networked control systems, delay compensation, real time systems

come

INTRODUCTION

Automation systems of the future and even those currently in use today, will consist of a large number of intelligent devices and control systems connected by local or global communication networks. In these Networked Control Systems (NCSs), communications between process, controllers, sensors and actuators are performed through the network.

In most cases, the insertion of the network does not significantly affect the performance of the control system. However, for some time constrained systems (constraints coming from the dynamic of the physical process to observe and to control), the implementation of the NCS should be done considering the implications introduced by the network. For such systems, the insertion of the communication network into the feedback control loop introduces an additional delay, either constant or time varying, that makes the analysis and the control design more complex.

Studying the networked control systems requires the evaluation of the Quality of Service level provided by the network and integration the values of these QoS parameters in the control design. One of the main limitations of the previous research on NCS is the lack of integrated control design and network evaluation. It is often assumed that either the control is specified (information timing constraints exists) or the performance of the network is well-defined (information about the delay distribution, uncertainty, deviation from mean value, missing value rate is assumed known). The results for the whole design cycle, where the network evaluation and control are integrated and performed the same context, are still lacking.

In this paper the whole design cycle for the NCS will be presented. A novel integrated approach will be presented where the estimated information about the network performance will be utilized in the control synthesis. First, the procedure of obtaining information about the delay (the upper bound delay) is presented and the obtained value is utilised in delay compensation. The delay algorithm presented applies ideas from network calculus theory, and the delay compensating strategies is based on robust control theory.

The paper is organized as follows: Chapter 2 is dedicated to introducing the upper bound delay estimation algorithm for the switched Ethernet network. In Chapter 3 the delay compensation strategy is introduced. Chapter 4 consists of the simulation results and discussion, and the paper ends with a concluding section in Chapter 5.

UPPER BOUND DELAY ESTIMATION

In the paper the switched Ethernet network is used as an example of the NCS network. The Ethernet networks are nowadays also more and more used in control applications and, in this context, it is important to understand the behaviour of the network in order to be able to control the network performance, such as delays [START_REF] Georges | Designing suitable switched Ethernet architectures regarding real-time application constraints[END_REF]. Next, the procedure of obtaining the upper bound delay over the network will be explained. The algorithm presented applies ideas from network calculus theory (see [START_REF] Cruz | A Calculus for Network Delay, Part I: Network Elements in Isolation[END_REF][START_REF] Boudec Le | Network Calculus, A Theory of Deterministic Queueing Systems for the Internet[END_REF][START_REF] Jasperneite | Deterministic Real-Time Communication with Switched Ethernet[END_REF]. For more details of the algorithm, see [START_REF] Georges | Confronting the performances of a switched Ethernet network with industrial constraints by using the Network Calculus[END_REF]. First, the method to obtain a maximum delay for crossing a single Ethernet switch will be explained in Section 2.1, and the procedure of obtaining end-toend delays in the network, based on the delays over the switches, will be given in Section 2.2.

Maximum delay for crossing the Ethernet switch

The first step in the Ethernet switch modelling is determination of an upper bound delay for crossing each of the basic components: FIFO multiplexer, FIFO queue, and demultiplexer. The upper bound delay over the switch is then the sum of the upper bound delays over these basic components, Figure 1:

output queue mux switch D D D D + + = (1)
where the notation D is used to represent the upper bound value of the delays. In the mathematical analysis, the traffic arriving at the switch, both periodic and aperiodic is modelled as a 'leaky bucket controller'. Data will arrive at the leaky rate only if the level of the bucket is less than the maximum bucket size. Traffic models are called the arrival curve in network calculus theory and, with the assumption that the traffic follows the leaky bucket controller and the incoming rate is limited by the port capacity, these curves are affine and have the form of:

) min( ) ( t t C t b in ρ σ + = (2)
where σ is the maximum amount of data that can arrive in a burst, ρ is an upper bound of the average rate of the traffic flow, and is the capacity of the input port. In the same way, service curves are used to represent the minimal data processing activity of the components. Typical arrival and service curves are shown in Figure 2.

in C
The approach is based on the evolution of a specific parameter, the backlog. The backlog is the number of bits waiting in the component, and it is a measure of congestion over the component. For the arrival curves in Figure 2, the upper bound backlog occurs at time t where the following line is a maximum:

t C C L t b t b out - + + ) / ( ) ( 2 2 1 (3)
where 1 b and are the arrival curves of stream 1 and 2 at time t, L is the maximum length of the frames, is the capacity of the import port 2, and is the capacity of the output link. When the upper bound backlog over the component is known, the upper bound delay over the component is then obtained by dividing the maximum backlog value by the capacity of the output link of the multiplexer. In a FIFO minputs multiplexer, the delay for any incoming bit from the stream i is upper-bounded by:

2 b 2 C out C k mux k out i mux B C D , , min 1 = (4)
where k mux B , is an upper-bound of the backlog in the bursty periods , such that . For k = i, the bursty period is defined by

k u m k ≤ ≤ 1 ) /( i i i i C u ρ σ - =
and the backlog is upper-bounded by :

( ∑ ≠ = - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = m i z z out i i z z i z z i mux C C u C L u B ; 1 , ρ σ ) (5)
where i σ is the burstiness of the stream i, i ρ is the average rate of arrival of the data of stream i, is the maximum length of the frames of stream i, and is the capacity of the import port i. For such that

i L i C i k ≠ m k ≤ ≤ 1 , we have k k k k k k C L C u / ) /( - - = ρ σ and
( )

k i i i m k z z out k k z z k z z i mux L C L C C u C L u B + - - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = ∑ ≠ = ρ ρ σ ; 1 , (6)
For the FIFO queue the delay of any byte is upperbounded by: ( )

in in in out in out queue C C C C D σ ρ - - = 1 (7)
For the demultiplexer it is assumed that the time required to route the output port is relatively negligible compared to the other delays, so that the demultiplexer does not generate delays.

Fig. 2. Arrival and service curves and backlog evolution inside the two-input FIFO multiplexer.

Maximum end-to-end delays for crossing a switched Ethernet network

The computation of the upper bound end-to-end delays requires that special attention is paid to the input parameters of the previous equations. Indeed, the maximum delay value D depends on the leaky bucket parameters: the maximum amount of traffic σ that can arrive in a burst and the upper bound of the average rate of the traffic flow ρ . In order to calculate the maximum delay over the network, it is hence necessary that the envelope ( ) ρ σ , is known at every point in the network. However, as shown in The last part of the method used to obtain the upperbounded delay estimate is the resolution of the burstiness characteristic of each flow at each point of the network. First, the burstiness values are determined by solving the equation system: where h is the number of crossed switches.

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ n n nn n n

COMPENSATING THE ESTIMATED TRANSMISSION NETWORK DELAY EFFECT WITH THE ROBUST CONTROL BASED APPROACH

In this chapter the information available about the dynamic behaviour of the network, the upper bound delay estimate, is integrated in the control law. The main idea behind the approach is first to represent the communication network induced delays in the frequency domain as an uncertainty around the nominal plant. Information about the upper bound delay is utilized in formulating the representation of the uncertainty. Next, robust control methods are used to generate a control law that is able to meet the design specifications, to maintain the system performance, and to make the system insensitive even in the worst case disturbance for all the plants in the uncertainty set. The "worst-case" uncertainty is assumed to occur when the network delay equals the theoretical upper bound delay estimated by the algorithm based on network calculus theory. In order to facilitate understanding, the synthesis of both is done in continuous time. For implementation the obtained controllers should be discretized.

Problem formulation

The problem is studied from the controller point of view as a control problem in which the properties of the communication network have to be taken into account in the controller design. For the controller, the system that is actually being controlled, the system P , is a combination of the nominal plant (assumed to be fixed and certain) and uncertain (unknown, but bounded) dynamical effects of the network E. With the weighting functions, , and normalized perturbations, , the following expression for the networked controlled plant is obtained:

G I w Δ P G 1 ) ( )); ( ) ( 1 )( ( ) ( 1 4 4 3 4 4 2 1 ≤ Δ ∞ ∀ ≤ Δ Δ + = I jw s s w s G s G I I p (12)

Representing the network induced delay.

Next, information about the upper bound delay of the network is utilized in creating the weighting functions . The weight can be obtained by finding the smallest radius that includes all possible plants: For example in this case the following weight can be chosen:

I w ) (w l I ) /( ; 2 ) /( ; 1 ) ( 1 max ) ( ) ( ) ( max ) ( 2 1 2 1 ) ( 2 1 2 1 2 1 ⎩ ⎨ ⎧ + > + < - = - = - = + Π ∈ Π ∈ UBD UBD
465 . 3 / ) ( 1 ) ( ) ( 2 1 2 1 s UBD UBD s UBD UBD s w I ⋅ + + ⋅ + = (15)
This delay estimate was originally proposed by [START_REF] Wang | Representation of uncertain time delays in the H infinity framework[END_REF] to represent uncertain delays in the framework in the design of robust controllers.

∞ H 3.3 Controller synthesis

For a SISO case the controller synthesis problem can be solved in a relatively straightforward manner, since the SISO case with one complex multiplicative perturbation the Robust Performance (RP) problem can be approximated as a weighted mixed sensitivity problem where the condition is slightly strengthened:

2 1 max 2 2 < + = ∞ T w S w T w S w I P I P ω (16)
Where is a weight for the sensitivity function S (usually an approximator of an integrator), and T is the complementary sensitivity function. For a MIMO case the use of a more complicated technique such as P w μ synthesis is required.

SIMULATION RESULTS AND DISCUSSIONS

In the simulations, the network of a real time process, a controller, and two overload traffic stations connected over a full duplex Ethernet switch, were used. The structure of the system is shown in Figure 4. To calculate the upper bound delay, the initial leaky bucket values of each stream were first identified. 6 are messages sent periodically. The traffic sent from the process to the controller is given by , and the traffic from the controller to the process by . The upper-bounds for these traffics will be computed in order to obtain the upper bounds, UBD

) ( 0 1 t b ) ( 0 2 t b
1 and UBD 2 . We consider also background traffic ( , , , ) from the stations to the process and to the controller in order to overload the network:

) ( 0 3 t b ) ( 0 4 t b ) ( 0 5 t b ) ( 0 6 t b Overload traffic 0 1 σ 1 2 σ 1 3 σ 1 4 σ 1 6 σ 1 1 σ 0 2
+ = + = = = = + = + = = ρ σ ρ σ (17)
Next, the route of each stream was identified and the output burstiness equations were formulated. After solving the burstiness values the end-to-end upper bound delay for streams 1 and 2 are:

ms UBD UBD 5 . 3 1 0 1 2 1 2 1 ≈ - = = ρ σ σ (18)
In evaluating the effects of the network on the control system performance, the following model of a real time process and a nominal controller were used (time in ms):

( )( ) 2 . 0 5 2 ) ( + + = s s s P 4529 . 0 , 5508 . 0 , ) ( = = + = I P I P K K s K s K s C (19)
The delay compensation strategy based on the robust control approach was implemented by solving the mixed sensitivity problem in Equation 16. Equation 15was used as a weighting function for the complementary sensitivity function T. As a weighting function for the sensitivity function S the following approximation of the integrator was implemented:

P w A s M s s w B B P ω ω + + = / ) ( (20)
Where B ω is the bandwidth where control is effective, M is the desired maximum peak of wb, and A is a small number used to avoid numerical problems. The optimal controller for this mixed sensitivity problem was found using the Matlab The simulation results are presented in Figures 567.

Figure 5 shows the evolution of the sensor and actuator network delays, Figure 6 represents the performance of the system when controlled with a the nominal controller (Equation 19) as the delay increases, and Figure 7 indicates the performance of the robust controller as the delay increases. From the figures it can be concluded that the nominal system becomes unstable when the delay increases. The stability of the feedback control loop is regained when a delay compensation strategy such as the robust control based approach is implemented.

CONCLUSIONS

In this paper an analysis of the networked control system has been presented. A procedure for obtaining the upper bound delay value in the switched Ethernet network was presented, and the obtained delay estimate was used in the control compensation. Two control compensation strategies, the Smith predictor based compensation strategy and the robust control based compensation strategy, were presented and compared. It can be concluded that the upper bound delay estimate is an important measure of the networked control system which can also be used for the design and synthesis of a control system.

Fig. 1

 1 Fig. 1 Model of a 2 port-switch in a full duplex mode based on shared memory and a cut-through management.

Figure 3 ,Fig. 3 .

 33 Figure 3, only the initial arrival curve values ( ) 0 0 , ρ σ are usually known, and the values for other arrival curves have to be determined. To calculate all the arrival curve values the following equations can be used:

  the above equation, the upper bound end-to-end delays are obtained from

  Fig. 4. The structure of the network

Fig. 7 .

 7 Fig. 5. Evolution of the sensor and actuator network delays.

ACKNOWLEDGEMENTS

This research has been conducted as a part of the Networked Control Systems Tolerant to Faults (NeCST) project IST-004303 that is partially funded by the EU. The authors gratefully acknowledge the support.


Christian Robert

Jean-Michel Marin
email: jean-michel.marin@inria.fr

On some difficulties with a posterior probability approximation technique

Keywords: Bayesian model choice, posterior approximation, reversible jump, Markov Chain Monte Carlo (MCMC), pseudo-priors, unbiasedness, improperty

 are biased and advance several arguments towards this thesis, the primary one being the confusion between model-based posteriors and joint pseudo-posteriors.

Introduction

Model selection is a fundamental statistical issue and a clear asset of the Bayesian methodology but it faces severe computational difficulties because of the requirement to explore simultaneously the parameter spaces of all models under comparison with enough of an accuracy to provide sufficient approximations to the posterior probabilities of all models. When [START_REF] Green | Reversible jump MCMC computation and Bayesian model determination[END_REF] introduced reversible jump techniques, it was perceived by the community as the second MCMC revolution in that it allowed for a valid and efficient exploration of the collection of models and the subsequent literature on the topic exploiting reversible jump MCMC is a testimony to the appeal of this method. Nonetheless, the implementation of reversible jump techniques in complex situations may face difficulties or at least inefficiencies of its own and, despite some recent advances in the devising of the jumps underlying reversible jump MCMC [START_REF] Brooks | Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions (with discussion)[END_REF], the care required in the construction of those jumps often acts as a deterrent from its applications.

There are practical alternatives to reversible jump MCMC when the number of models under consideration is small enough to allow for a complete exploration of those models. Integral approximations using importance sampling techniques like those found in [START_REF] Gelfand | Bayesian model choice: asymptotics and exact calculations[END_REF], based 1 on an harmonic mean representation of the marginal densities, and in [START_REF] Gelman | Simulating normalizing constants: From importance sampling to bridge sampling to path sampling[END_REF], focussing on the optimised selection of the importance function, are advocated as potential solutions, see [START_REF] Chen | Monte Carlo Methods in Bayesian Computation[END_REF] for a detailed entry. The reassessment of those methods by [START_REF] Bartolucci | Efficient Bayes factor estimation from the reversible jump output[END_REF] showed the connection between a virtual reversible jump MCMC and importance sampling (see also [START_REF] Chopin | Contemplating evidence: properties, extensions of, and alternatives to nested sampling[END_REF]. In particular, those papers demonstrated that the output of MCMC samplers on each single model could be used to produce approximations of posterior probabilities of those models, via some importance sampling methodologies also related to [START_REF] Newton | Approximate Bayesian inference by the weighted likelihood boostrap (with discussion)[END_REF].

In [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], a new and straightforward method is advanced to compute posterior probabilities of models under scrutinity based solely on MCMC outputs restricted to single models. While this simplicity is quite appealing for the approximation of those probabilities, we believe that both proposals of [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] are inherently biased and we advance in this note several arguments towards this thesis. In addition, we notice that, to overcome the bias we thus exhibited, a valid solution would call for the joint simulation of parameters under all models (using priors or pseudo-priors), unless the alternative advanced by [START_REF] Green | Model choice with mcmc on product spaces without using pseudo-priors[END_REF] is is used, and this step would thus loose the primary appeal of the methods against the one proposed by [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF], from which both [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] are inspired.

The methods

In a Bayesian framework of model comparison (see, e.g., [START_REF] Robert | The Bayesian Choice[END_REF], given D models in competition, M k , with densities f k (y|θ k ), and prior probabilities ̺ k = P (M = k) (k = 1, . . . , D), the posterior probabilities of the models M k conditional on the data y are given by

P (M = k|y) ∝ ̺ k f k (y|θ k )π k (θ k ) dθ k ,
the proportionality term being given by the sum of the above and M denoting the unknown model index.

In the specific setup of hidden Markov models, the solution of Scott (2002, Section 4.1) is to generate simultaneously and independently D MCMC chains (θ

(t) k ) t , 1 ≤ k ≤ D ,
with stationary distributions π k (θ k |y) and to approximate P (M = k|y) by

̺k (y) ∝ ̺ k T t=1 f k (y|θ (t) k ) D j=1 ̺ j f j (y|θ (t) j ) ,
as reported in formula (21) of [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF], with the mention that "(21) averages the D likelihoods corresponding to each θ j over the life of the Gibbs sampler" (p.347), the later being understood as "independently sampled D parallel Gibbs samplers" (p.347).

From a more general perspective, the proposal of [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] for an approximation of the P (M = k|y)'s follows both from [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] approximation and from the pseudo-prior construction of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF] that predated reversible jump MCMC by saturating the parameter space with an artificial simulation of all parameters at each iteration. However, due to a very special (and, we believe, mistaken) choice of pseudo-priors discussed below, Congdon's (2006, p.349) approximation of P (M = k|y) eventually reduces to the estimator

̺k (y) ∝ ̺ k T t=1 f k (y|θ (t) k )π k (θ (t) k ) D j=1 ̺ j f j (y|θ (t) j )π j (θ (t) j ) ,
where the θ (t) k 's are samples from π k (θ k |y) (or approximate samples obtained by an MCMC algorithm). (A very similar proposal is found in [START_REF] Congdon | Model weights for model choice and averaging[END_REF] and, while some issues like the use of improper pseudo-priors discussed in Section 3.3 are corrected, the fundamental difficulty of simulating from the wrong target remains. We thus chose to address primarily the initial paper by [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], especially since it generated follow-up papers like [START_REF] Chen | Bayesian model selection for heteroskedastic models[END_REF] and since its inherent simplicity is likely to appeal to unwary readers.)

Although both approximations ̺k (y) and ̺k (y) differ in their expressions, they fundamentally relate to the same notion that parameters from other models can be ignored when conditioning on the model index M . This approach is therefore bypassing the simultaneous exploration of several parameters spaces and restricts the simulation to marginal samplers on each separate model. This feature is very appealing since it cuts most of the complexity from the schemes both of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF] and of [START_REF] Green | Reversible jump MCMC computation and Bayesian model determination[END_REF]. We however question the foundations of those approximations as presented in both [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF][START_REF] Congdon | Model weights for model choice and averaging[END_REF] and advance below arguments that both authors are using incompatible versions of joint distributions on the collection of parameters that jeopardise the validity of the approximations.

Difficulties

The sections below expose the difficulties found with both methods, following the points made in [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], respectively. The fundamental difficulty with their approaches appears to us to be related to a confusion between the model dependent simulations and the joint simulations based on a pseudo-prior scheme as in [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF]. Once this difficulty is resolved, it appears that the approximation of P (M = k|y) by P (M = k|y) does require a joint simulation of all parameters and thus that the solutions proposed in [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] are of the same complexity as the proposal of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF].

Incorrect marginals

We denote by θ = (θ 1 , . . . , θ D ) the collection of parameters for all models under consideration. Both [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] (1)

∝ D k=1 ̺ k f k (y|θ k ) j π j (θ j ) = D k=1 ̺ k m k (y) π k (θ k |y) j =k π j (θ j ) . (2)
In both papers, the θ (t) 's are instead simulated as independent outputs from the componentwise posteriors π k (θ k |y) and this divergence jeopardises the validity of the approximation. The error in their interpretations stems from the fact that, while the θ

(t)
k 's are (correctly) independent given the model index M , this independence does not hold once M is integrated out, which is the case in the above approximation P (M = k|y).

MCMC versus marginal MCMC

When [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] defines a Markov chain (θ (t) ) at the top of page 349, he indicates that the components of θ (t) are made of independent Markov chains (θ (t) k ) simulated with MCMC samplers related to the respective marginal posteriors π k (θ k |y), following the approach of [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF]. The aggregated chain (θ (t) ) is thus stationary against the product of those marginals,

D k=1 π k (θ k |y) .
However, in the derivation of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF], the model is defined in terms of (1) and the Markov chain should thus be constructed against (1), not against the product of the model marginals. Obviously, in the case of [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], the fact that the pseudo-joint distribution does not exist because of the flat prior assumption (see Section 3.3) prevents this construction but, in the event the flat prior is replaced with a proper (pseudo-) prior (as in [START_REF] Congdon | Model weights for model choice and averaging[END_REF], the same statement holds: the probabilistic derivation of P (M = k|y) relies on the pseudo-prior construction and, to be valid, it does require the completion step at the core of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF], where parameters need to be simulated from the pseudo-priors.

Similarly, in [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF], the target of the Markov chain (θ (t) , M (t) ) should be the distribution

P (θ, M = k|y) ∝ π k (θ k ) ̺ k f k (y|θ k ) j =k π j (θ j )
and the θ (t) j 's should thus be generated from the prior π j (θ j ) when M (t) = j-or equivalently from the corresponding marginal if one does not condition on M (t) , but simulating a Markov chain with stationary distribution ( 2) is certainly a challenge in many settings if the latent variable decomposing the sum is not to be used.

Since, in both [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] and [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], the (θ (t) )'s are not simulated against the correct target, the resulting averages of P (M = k|y, θ (t) ), ̺k (y) and ̺k (y), will both be biased, as demonstrated in the example of Section 3.4.

Improperty of the posterior

When resorting to the construction of pseudo-posteriors adopted by [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF], [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] uses a flat prior as pseudo-prior on the parameters that are not in model M k . More precisely, the joint prior distribution on (θ, M ) is given by [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] formula (2),

P (θ, M = k) = π k (θ k ) ̺ k j =k π(θ j |M = k) = π k (θ k ) ̺ k ,
which is indeed equivalent to assuming a flat prior as pseudo-prior on the parameters θ j that are not in model M k .

Unfortunately, this simplifying assumption has a dramatic consequence in that the corresponding joint posterior distribution of θ is never defined (as a probability distribution) since

π(θ|y) = D k=1 π k (θ k |y) P (M = k|y)
does not integrate to a finite value in any of the θ k 's (unless their support is compact). When [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] points out "that it is not essential that the priors for P (θ j =k |M = k) are improper" (p.348), the whole issue is that they cannot be improper. (An alternative is to implement the jumping scheme of [START_REF] Green | Model choice with mcmc on product spaces without using pseudo-priors[END_REF], in which case pseudo-priors become irrelevant, but the estimator of P (θ, M = k) then reduces to [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] ̺k .)

The fact that the posterior distribution on the saturated vector θ = (θ 1 , . . . , θ D ) does not exist obviously has dire consequences on the subsequent derivations, since a positive recurrent Markov chain with stationary distribution π(θ|y) cannot be constructed. Similarly, the fact that P (M = k|y) = P (θ, M = k|Y ) dθ does not hold any longer.

Note that [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] does not follow the same track: when defining the pseudo-priors in his formula (20), he uses the product definition1

P (θ, M = k) = π k (θ k ) ̺ k j =k π j (θ j ) ,
which (seemingly) means that the true priors are also used as pseudo-priors across all models. However, we stress that [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] does not refer to the construction of [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF] in his proposal.

Illustration

We now proceed through a toy example where all posterior quantities can be computed in order to evaluate the bias brought by both approximations.

Example 1. Consider the case when a model M 1 : y|θ ∼ U(0, θ) with a prior θ ∼ Exp( 1) is opposed to a model M 2 : y|θ ∼ Exp(θ) with a prior θ ∼ Exp(1). We also assume equal prior weights on both models: ̺ 1 = ̺ 2 = 0.5.

The marginals are then

m 1 (y) = ∞ y θ -1 e -θ dθ = E 1 (y) ,
where E 1 denotes the exponential integral function tabulated both in Mathematica and in the GSL library, and

m 2 (y) = ∞ 0 θe -θ(y+1) dθ = 1 (1 + y) 2 .
For instance, when y = 0.2, the posterior probability of M 1 is thus equal to

P (M = 1|y) = m 1 (y)/{m 1 (y) + m 2 (y)} = E 1 (y)/{E 1 (y) + (1 + y) -2 } ≈ 0.6378 ,
while, for y = 0.9, it is approximately 0.4843. This means that, in the former case, the Bayes factor of M 1 against M 2 is B 12 ≈ 1.760, while for the later, it decreases to B 12 ≈ 0.939.

The posterior on θ in model M 2 is a gamma Ga(2, 1+y) distribution and it can thus be simulated directly. For model M 1 , the posterior is proportional to θ -1 exp(-θ) for θ larger than y and it can be simulated using a standard accept-reject algorithm based on an exponential Exp(1) proposal translated by y.

Using simulations from the true (marginal) posteriors and the approximation of [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF], the numerical value of ̺1 (y) based on 10 6 simulations is 0.7919 when y = 0.2 and 0.5633 when y = 0.9, which translates into Bayes factors of 3.805 and of 1.288, respectively. For the approximation of [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF], the numerical value of ̺1 (y) is 0.6554 (corresponding to a Bayes factor of 1.898) when y = 0.2 and 0.6789 when y = 0.9 (corresponding to a Bayes factor of 2.11), based on the same simulations. Note that in the case y = 0.9, a selection based on either approximation of the Bayes factor would select the wrong model.

If we use instead a correct simulation from the joint posterior (2), which can be achieved by using a Gibbs scheme with target distribution P (θ, M = k|y), we then get a proper MCMC approximation to the posterior probabilities by the P (M = k|y)'s. For instance, based on 10 6 simulations, the numerical value of P (M = 1|y) when y = 0.2 is 0.6370, while, for y = 0.9, it is 0.4843. Note that, due to the impropriety difficulty exposed in Section 3.3, the equivalent correction for [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] scheme cannot be implemented.

In Figure 1, the three approximations are compared to the exact value of P (M = 1|y) for a range of values of y. The correct simulation produces a graph that is indistinguishable from the true probability, while [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] approximation stays within a reasonable range of the true value and [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] surprisingly drifts apart for most values of y. ◭ The correspondence of what is essentially [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF] scheme with the true numerical value of the posterior probabilities is obviously unsurprising in this toy example but more advanced setups see the approximation degenerate, since the simulations from the prior are most often inefficient, especially when the number of models under comparison is large. This is the reason why [START_REF] Carlin | model choice through Markov chain Monte Carlo[END_REF] introduced pseudo-priors that were closer approximations to the true posteriors. 2002) approximation (in blue), [START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] approximation (in green), and correction of [START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] approximation (in red), indistinguishable from the true value (based on N = 10 6 simulations).

  start from the representation P (M = k|y) = P (M = k|y, θ)π(θ|y) dθ to justify the approximation P (M = k|y) = T t=1 P (M = k|y, θ (t) )/T . 3 This is indeed an unbiased estimator of P (M = k|y) provided the θ (t) 's are generated from the correct (marginal) posterior π(θ|y) = D k=1 P (θ, M = k|y)

Figure 1 :

 1 Figure 1: Comparison of three approximations of P (M = 1|y) with the true value (in black): Scott's (2002) approximation (in blue),[START_REF] Congdon | Bayesian model choice based on Monte Carlo estimates of posterior model probabilities[END_REF] approximation (in green), and correction of[START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] approximation (in red), indistinguishable from the true value (based on N = 10 6 simulations).

The indices on the priors have been added to make notations coherent, since[START_REF] Scott | Bayesian methods for hidden Markov models: recursive computing in the 21st Century[END_REF] denotes all priors with the same letter p.

Acknowledgements

Both authors are grateful to Brad Carlin for helpful discussions and to Antonietta Mira for providing a perfect setting for this work during the ISBA-IMS "MCMC'ski 2" conference in Bormio, Italy. The second author is also grateful to Kerrie Mengersen for her invitation to "Spring Bayes 2007" in Coolangatta, Australia, that started our reassessment of those papers. This work had been supported by the Agence Nationale de la Recherche (ANR, 212, rue de Bercy 75012 Paris) through the 2006-2008 project Adap'MC.