Lou Lee
email: lou.lee@univ-lorraine.fr

Denis Jouvet
email: denis.jouvet@loria.fr

Katarina Bartkova
email: katarina.bartkova@univ-lorraine.fr

Yvon Keromnes
email: yvon.keromnes@univ-lorraine.fr

Mathilde Dargnat
email: mathilde.dargnat@univ-lorraine.fr

Correlation between prosody and pragmatics: case study of discourse markers in French and English

Keywords: prosody, pragmatics, discourse markers, F0 patterns

published or not. The documents may come

Introduction

Discourse markers are words that are used on a discursive or situational level instead of a referential or textual level. They convey a different pragmatic meaning depending on the context or discourse situation. There have been a great number of empirical studies wholly dedicated to this semantico-pragmatic phenomenon for the past decades (e.g., [START_REF] Hansen | The function of discourse particles: A study with special reference to spoken standard French[END_REF], [START_REF] Schiffrin | Discourse markers[END_REF], [START_REF] Aijmer | English discourse particles: Evidence from a corpus[END_REF]). In order to understand them accurately, it is essential to study their pragmatic functions, that is the communicative goal they serve in discourse situations. Some authors have studied different pragmatic functions of discourse markers and the ambiguity in their meanings as well as their disambiguation [START_REF] Hirschberg | Empirical studies on the disambiguation of cue phrases[END_REF]. The multifunctionality and ambiguity of discourse markers often stem from their dependency on the context to define their meaning [START_REF] Cuenca | Pragmatic markers in contrast: The case of well[END_REF], [START_REF] Chanet | 1700 occurrences de la particule quoi en français parlé contemporain: approche de la «distribution» et des fonctions en discours[END_REF]. In fact, not only discourse markers can have an almost infinite number of functions depending on context, but also different markers can share some of their meanings [START_REF] Aijmer | Pragmatic markers[END_REF].

Prosody is frequently used in research on discourse or emotional speech due to the fact that it expresses a meaning going beyond the traditional semantic meaning [START_REF] Tseng | Higher level organization and discourse prosody[END_REF], [START_REF] Liebenthal | The language, tone and prosody of emotions: neural substrates and dynamics of spoken-word emotion perception[END_REF], [START_REF] Sauter | Perceptual cues in nonverbal vocal expressions of emotion[END_REF]. However, the prosody of discourse markers has been relatively little studied, especially for the French language. Prosody can convey very important and useful information allowing to detect discursive and pragmatic meanings of these markers.

A few recent studies have found correlation between prosodic parameters of discourse markers and their usage [START_REF] Horne | Discourse markers and the segmentation of spontaneous speech-The case of Swedish men 'but/and/so[END_REF], [START_REF] Wichmann | Prosody and discourse: a diachronic approach[END_REF], [START_REF] Cabarrão | Prosodic classification of discourse markers[END_REF]. According to these studies, prosodic information predicts successfully whether the word is used as a marker or a non-marker. For instance, discourse markers are often phonetically reduced or unstressed compared to the 'standard' usage of the same word not as a marker [START_REF] Brinton | The evolution of pragmatic markers in English: Pathways of change[END_REF]. In addition, prosody can predict more specific pragmatic meanings of the marker in context. There are also a number of case studies on a specific discourse marker, on how prosody reflects this semantico-pragmatic change in a word [START_REF] Wichmann | How prosody reflects semantic change: A synchronic case study of of course[END_REF], [START_REF] Gravano | On the role of context and prosody in the interpretation of 'okay[END_REF], [START_REF] Benus | Prosody, emotions, and…'whatever[END_REF].

In this study, we present a prosodic analysis of three French discourse markers (alors ['so', 'then', 'well'], bon ['well', 'right'], and donc ['so', 'then']) and three English markers (now, so, and well) according to their different pragmatic functions. As we observed in our previous studies on French discourse markers [START_REF] Dargnat | Discourse particles in French: prosodic parameters extraction and analysis[END_REF], [START_REF] Bartkova | How to be a Discourse Particle?[END_REF], [START_REF] Jouvet | Analysis and automatic classification of some discourse particles on a large set of French spoken corpora[END_REF], [START_REF] Lee | Can prosody meet pragmatics? Case of discourse particles in French[END_REF], prosodic production of discourse markers depends on their pragmatic function in the context. The present study is focussed on a comparison between prosodic correlates of discourse markers with respect to pragmatic functions, firstly in different speech styles and secondly across different languages. The first comparison is conducted on French speech data between spontaneous and prepared speech. Then another comparison is conducted on prepared speech between French and English data.

The paper is organised as follows. Section 2 presents the speech corpus, the annotation of the data and the prosodic features used. Section 3 compares the prosodic characteristics of the words under consideration between two speech styles (prepared vs. spontaneous) on the French data. Section 4 presents a comparison between French and English data on prepared speech, followed by the conclusion of the study.

Speech corpora and annotations

French and English corpora

Occurrences of the words studied, frequently used as discourse markers, were randomly extracted from French and English corpora. The French corpora cover various speech styles of different levels of spontaneity: ESTER2 [START_REF] Galliano | The ESTER 2 evaluation campaign for the rich transcription of French radio broadcasts[END_REF], a corpus of radio broadcast news composed of prepared speech by journalists and some interviews, and the corpora of the project ORFEO [START_REF] Debaisieux | Le projet ORFEO : Un corpus d'études pour le français contemporain[END_REF] composed of a large variety of speech from storytelling to interviews and daily dialogues. They amount for several hundreds of hours of speech data. For English, the corpus TED-LIUM 3 [START_REF] Hernandez | TED-LIUM 3: twice as much data and corpus repartition for experiments on speaker adaptation[END_REF] was used. It is composed of 452 hours of speech, recordings of TED Talks. Manual transcriptions of these corpora were automatically aligned with the speech signal allowing us to locate the positions of the words.

As already stated, the words under study are: alors, bon, donc for French, and now, so, well for English. These words, often used as discourse markers, were chosen because they are frequent in our corpora, and thus we can expect to have enough data for a relevant and reliable analysis. For French, 500 to 1,000 occurrences were randomly extracted for each word studied, with a context of 15 words before and 15 words after so that it is possible to understand the meaning and function of the word in its context. Each word occurrence was annotated manually with labels indicating its usage: discourse marker (DM) or not (non-DM). The annotation was further refined for the discourse markers to determine their pragmatic functions. For most of the words, approximately 70% of their occurrences were identified as discourse markers. The same process was conducted for the English data, but only 200 occurrences were extracted for each word studied. The phonemic segmentation of the words of interest (i.e., the word under study, and its immediate left and right context) was manually verified and corrected where necessary.

Pragmatic functions of discourse markers

The semantic load of a word when used as a discourse marker is significantly lighter compared to the one when used as a nonmarker [START_REF] Hansen | The function of discourse particles: A study with special reference to spoken standard French[END_REF], [START_REF] Maschler | Discourse markers: Language, meaning, and context[END_REF]. A discourse marker conveys mainly pragmatic meanings in the discourse. Besides, more than one pragmatic meaning can be attributed to a discourse marker, depending on its context.

For each word under study, pragmatic function labels were chosen first according to the literature [START_REF] Beeching | La co-variation des marqueurs discursifs bon, c'est-à-dire, enfin, hein, quand même, quoi et si vous voulez : une question d'identité ?[END_REF], [START_REF] Degand | Alors between discourse and grammar: the role of syntactic position[END_REF], [START_REF] Denturck | Étude des marqueurs discursifs. L'exemple de quoi[END_REF], [START_REF] Lefeuvre | Bon et quoi à l'oral : marqueurs d'ouverture et de fermeture d'unités syntaxiques à l'oral[END_REF], then were adjusted during the process of annotation to cover other pragmatic functions observed in our own speech data. Each discourse marker under study has from three up to seven different pragmatic functions, and these functions can be shared among different discourse markers (cf. [START_REF] Lee | Can prosody meet pragmatics? Case of discourse particles in French[END_REF] for more information on the annotation process). Table 1 shows the number of occurrences that are used as discourse markers, and the frequency of each pragmatic function for each discourse marker. The table excludes infrequent labels and complex discourse marker usages (when more than one marker is used in combination, for instance "enfin bon"), which explains why the sum of the percentages in the table does not reach 100%. This is particularly the case for "bon" which is frequently used with other markers, probably due to its monosyllabicity.

In the remainder of the paper, we will limit the analysis to pragmatic functions that are shared by several discourse markers and appear frequently in both languages.

Prosodic parameters in question

In the following we will analyse the presence of pauses before or after the discourse markers, as well as the evolution of the fundamental frequency.

The presence of pauses before and/or after the discourse markers is determined from the phonetic alignments, that were obtained automatically, checked manually and corrected when necessary, with respect to the studied discourse markers and their immediate left and right contexts.

The fundamental frequency values (F0) have been computed using the RAPT algorithm [START_REF] Talkin | A robust algorithm for pitch tracking (RAPT)[END_REF] from the SPTK toolkit [START_REF] Sptk | [END_REF], which provides good F0 estimates in many conditions [START_REF] Jouvet | Performance Analysis of Several Pitch Detection Algorithms on Simulated and Real Noisy Speech Data[END_REF]. The F0 values have been converted into semi-tones, in comparison to the median F0 value of each speaker [START_REF] De Looze | L'échelle OME (Octave-MEdiane) : une échelle naturelle pour la mélodie de la parole[END_REF]. This allows a better interpretation of the F0 values with respect to the speaker's theoretical vocal range, which extends from the median F0 value minus 6 semi-tones (half an octave below) to the median value plus 6 or 12 semi-tones (half an octave or an octave above). We analysed the F0 movements between the words under consideration and the stressed or potentially stressed vowels of their immediate contexts (i.e., previous and following words). In French, we have considered the F0 value on the last (potentially stressed) vowel of the left context (w-1), on the last (potentially stressed) vowel of the word under consideration, and on the last (potentially stressed) vowel of the following word (w+1). In English, we have considered the F0 values on the stressed vowel of the left context (w-1), on the vowel of the word under consideration (all the English markers studied are monosyllabic), and on the stressed vowel of the following word (w+1). A vector quantisation procedure was then applied to all the data to extract representative patterns of F0 movements. Furthermore, patterns of difference in value of under 3 semi-tones were merged. Then the patterns were interpreted in terms of F0 movements (rising, plateau or falling) depending on whether the difference in the F0 values was larger than 3 semi-tones or not [START_REF] Mertens | Pitch contour stylization using a tonal perception model[END_REF]. This led to a set of 15 patterns that are later denoted according to the F0 movements, and to the F0 level on the discourse markers. For example, 'r_H_p' indicate a high F0 level on the word in question, rising from the left context, and plateau toward the right context. F0 values for frequently used patterns are reported in Table 3. In the examples reported, the notations for the slopes are 'rr' when rising more than 6 semi-tones, 'r' when rising more than 3 semi-tones, 'p' when it is a plateau and 'f' when falling. The F0 level on the words in question is denoted 'H' for high (more than 3 semi-tones above median value), 'Mh' for medium-high ( about 1 semi-tone above median value), 'M' for median value, and 'L' for low, (lower than 3 semi-tones beneath median value).

Impact of speaking style

The first set of comparisons concerns the behaviour of the prosodic parameters according to the speaking style, that is taneous speech vs. prepared speech in French.

Pauses before and/or after discourse markers

The presence of pauses immediately before and after the words under study is detailed in Table 2. The table shows the number of occurrences of the words used as a non-marker, and then those of each pragmatic function when used as a discourse marker. Regarding the presence of pauses, there are four possible contexts: no pause (neither before nor after), a pause before only, a pause after only, and a pause before and after.

The first observation we can draw from the table is the difference of pause occurrences between the usage of the word as non-DM (i.e., with its original lexical meaning) and the usage of the word as a discourse marker. Non-DM are most frequently accompanied by an absence of pause, both before and after the word.

When the words are used as discourse markers, for most of the cases, there are frequent occurrences of pauses, mainly before the discourse marker, and to a smaller extent before and after the discourse marker. Overall, pause occurrences are quite similar between spontaneous speech and prepared speech for the discourse markers 'alors' and 'bon'. 'bon' was never used to introduce a conclusion in prepared speech, whereas it was used for that in spontaneous speech with a pause before, and also after it. However, significant differences are observed for the discourse marker 'donc'. For the pragmatic function 'Introduction', 'donc' is almost always produced without a pause in prepared speech, whereas a pause before is the prevalent case in spontaneous speech. We observe a somewhat similar behaviour for the pragmatic functions 'Conclusion' and 'Parenthesis', but the difference in the presence of pauses before, between prepared and spontaneous speech, is much smaller.

F0 articulation with left and right contexts

Table 4 reports the frequency of the most frequently observed F0 patterns on French spontaneous and French prepared speech and English prepared speech, when there is no pause, neither before, nor after the word. The F0 values of the patterns are reported in Table 3. 'Falling-median-plateau', 'rising-median-plateau' and 'plateau-median' are frequent patterns for French words when not used as discourse markers, and even more frequent in spontaneous speech.

For the French language, and for the pragmatic function 'Introduction', the frequency of the F0 patterns is similar in spontaneous speech and prepared speech. This is almost the same for 'Conclusion' if we accumulate the frequencies of occurrences of the 'falling-median-plateau' and 'rising-median-plateau' patterns. For 'Parenthesis', the 'plateau-median-plateau' pattern is more frequently used in spontaneous speech.

Due to lack of space, we will not report results on patterns observed when there is a pause before, and when only the F0 articulation with to the right context is considered.

Comparison between the two languages

In this section, we compare the prosodic characteristics of our discourse markers in French in English.

Pauses before and/or after discourse markers

Table 2 shows no significant differences between the English and the French words when the words are not used as discourse markers. In this case, most of the time, there is no pause, neither before, nor after the words under consideration.

When the English words are used as discourse markers, the presence of a pause before becomes the dominant pattern for all the pragmatic functions considered. And in a significant number of cases there is also a pause after the discourse markers. Overall the pause occurrences for the three English discourse markers are rather similar to what is observed for the French discourse markers 'alors' and 'bon'.

The frequent presence of pauses around the discourse markers studied in both languages shows their relative syntactic independence highlighted by prosodic means. Indeed, when the semantic load of these words is maintained, they are more prosodically anchored, occurring in non-separated intonation groups whereas they frequently occur in independent intonation groups when used as discourse markers.

F0 movements with respect to immediate context

Comparison between F0 patterns in French and English prepared speech is reported in the last two columns of Table 4.

When the word is not used as a discourse marker, median F0 level is frequently observed in both languages.

When a discourse marker has the pragmatic function 'Introduction', the pattern 'rising-High-falling' is very frequent in English, whereas it is less frequent in French. For 'Conclusion' and 'Parenthesis', the 'plateau-median-high-plateau' pattern is almost never used in French data, but frequently used in English. A similar case is observed for the 'plateau-Median-risingrising' pattern for the pragmatic function 'Parenthesis'.

Conclusions

In this paper, we have analysed the prosodic characteristics of 6 words frequently used as discourse particles in French and in English. Prosodic characteristics are compared between two speaking styles, spontaneous speech vs. prepared speech, on the French data. Moreover, on prepared speech, prosodic characteristics are compared between French and English discourse markers. We have focused our analysis on the presence of pauses before and/or after the words under consideration, and on the F0 patterns and their articulation with the immediate, left and right contexts. The analysis has been carried out on several hundred of occurrences that have been randomly extracted from large speech corpora.

With respect to the presence of pauses around the words, the analysis shows significant differences between words keeping their lexical meaning (i.e. not discourse markers) with very few occurrences of pauses observed, and words used as discourse markers where pauses are frequently observed before and also sometimes after the words. This is true in both speaking styles and in both languages. Some differences in pause occurrences are observed between spontaneous speech and prepared speech, but differences are smaller in prepared speech across the two languages F0 patterns with respect to immediate left and right contexts have also been analysed and compared between speaking styles and languages. The usage of some patterns appears to be language specific, although many patterns are frequently used in both languages.

Acknowledgements

This work has been partially supported by the CPER LCHN (Contrat Plan État Région "Langues, Connaissances et Humanités Numériques").

  The following are the pragmatic functions considered in this study, with associated examples ({transcription, where '+' indicates the presence of a pause}, "file ID", and [time interval]):

• Introduction (to introduce a new topic or start talking): {that's what color printers do i'll show you in a moment that's actually quite useful ++ now the next question that you would like to ask is what would you like to} "George-Whitesides_2009X" [386.955-395.960] • Conclusion (to introduce a conclusion): {since been found in countless other + experiments around the globe from the us + to india +++ so + here's what i've learned ++ when it comes to poverty + we + the rich + should stop} "Rutger-Bregman_2017" [573.910-591.840] • Parenthesis (to add a comment or an extra information): {like ok let's look at television so they looked at an episode of lost now i don't have a tv which makes me a freak but very} "Jenny8Lee_2008P" [744.995-749.985] • Resuming (to continue a topic after a detour or a pause): {i've not yet done anything + to make any human being remember that i have lived ++ so fueled by that ambition he returned to the state legislature he eventually won a} "DorisKearnsGoodwin_2008" [216.735-226.345]

Table 1 :

 1 Number of occurrences of discourse markers, and percentage of associated pragmatic function

		French-spontaneous		French-prepared			English-prepared
	Pragmatic function	alors	bon	donc	alors	bon	donc	now	so	well
	Number of items	668	570	618	125	106	161	76	109	108
	Introduction	12%	6%	6%	34%	20%	10%	33%	12%	6%
	Resuming	15%		22%	3%		20%	13%	27%	9%
	Conclusion	14%	4%	23%	8%	0%	21%		39%
	Hesitation	6%	7%	9%	4%	8%	3%
	Parenthesis	15%	17%	18%	32%	26%	31%	54%	21%	24%
	Reformulation	3%	8%	4%	2%	9%	7%			5%
	Confirmation		10%			17%
	Closing		6%			8%
	Quotation								1%	23%
	Self-response									26%
	Turn-taking	11%			10%					6%

Table 3 :

 3 F0

		values (compared to speaker's median F0
		value) for frequently obtained F0 patterns
		No. of items F0 on w-1	F0 on w	F0 on w+1
	rr-H-p	63	-1.2	7.2	5.3
	r-H-f	135	-0.8	3.6	0.6
	p-Mh-p	137	1.7	1.2	3.7
	f-M-p	215	3.6	0.1	0.1
	r-M-p	133	-4.9	-0.4	-0.5
	p-M-p	388	-0.3	-0.5	-0.3
	p-M-rr	69	-1.6	-0.8	8.7
	p-L-r	136	-2.5	-4.0	0.9

Table 2 :

 2 Number of occurrences of discourse markers and percentage of position of pause(s) in the immediate context

			French-spontaneous	French-prepared	English-prepared
	Pragmatic function Position of pause(s)	alors	bon	donc	alors	bon	donc	now	so	well
		No. of items	78	92	19	57	72	18	98	54	67
	None	none	58%	68%	53%	46%	75%	67%	59%	78%	70%
	(Non-DM usage)	before only	37%	5%	21%	37%	6%	6%	8%	15%	3%
		before and after	0%	0%	0%	9%	4%	17%	2%	0%	0%
		No. of items	69	33	32	42	21	16	25	13	7
	Introduction	none before only	13% 57%	30% 48%	41% 56%	10% 69%	29% 67%	94% 6%	12% 52%	15% 46%	43% 57%
		before and after	29%	6%	3%	19%	5%	0%	32%	31%	0%
		No. of items	100		136	4		32	10	28	10
	Resuming	none before only	12% 73%		29% 65%	0% 75%		31% 53%	0% 70%	4% 61%	0% 60%
		before and after	13%		5%	25%		12%	30%	32%	40%
		No. of items	92	20	139	10	0	33		41
	Conclusion	none before only	34% 36%	10% 50%	42% 50%	30% 50%	0% 0%	55% 39%		24% 61%
		before and after	10%	40%	4%	20%	0%	0%		12%
		No. of items	101	97	110	40	28	50	41	23	26
	Parenthesis	none before only	36% 54%	54% 36%	57% 38%	42% 40%	39% 32%	84% 8%	20% 54%	43% 48%	15% 62%
		before and after	8%	8%	2%	10%	21%	0%	24%	4%	15%

Table 4 :

 4 Number of occurrences of discourse markers and percentage of most frequent patterns

NoItems rr_H_p r_H_f p_Mh_p f_M_p r_M_p p_M_p f_M_p _M_rr p_Ml_p p_L_r None (Non-DM usage)

		French-Spontaneous	181	2%	13%	8%	10%	6%	25%	3%	4%	7%	6%
		French-Prepared	92 10%	11%	5%	16%	4%	11%		7%	4%	4%
		English-Prepared	147	7%	10%	12%	7%	3%	19%	5%	1%	7%	5%
		French-Spontaneous	28		11%	11%	14%	7%	25%			7%	11%
	Introduction	French-Prepared	25	4%	12%	4%	16%	4%	24%	12%	4%	4%	4%
		English-Prepared	7 14%	29%	14%			14%		14%		14%
		French-Spontaneous	88		3%	3%	14%	9%	24%	2%	1%	15%	12%
	Conclusion	French-Prepared	22	5%	9%	5%	23%		18%		5%	14%	9%
		English-Prepared	12	8%		25%		25%	42%
		French-Spontaneous	193	1%	5%	3%	19%	11%	25%	8%	3%	7%	7%
	Parenthesis	French-Prepared	70	1%	10%	6%	24%	6%	11%	6%	3%	7%	9%
		English-Prepared	22	5%	5%	18%	9%	18%	9%	5%	18%		9%


Pierre Druilhet
email: pierre.druilhet@uca.fr

Erwan Saint

Loubert Bié

Improper vs finitely additive distributions as limits of countably additive probabilities

Keywords: bayesian statistics, improper distribution, finitely additive probability, q-vague convergence, uniform distribution, expected logarithmic convergence., remote probability

The Bayesian paradigm with proper priors can be extended either to improper distributions or to finitely additive probabilities (FAPs). Specific convergence modes are associated to improper distributions and to FAPs, so that, improper distributions and diffuse FAPs can be seen as limits of proper distribution sequences. In this paper, we compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct features of the limit behavior of a sequence of proper distribution. More specifically, an improper distribution characterizes the behavior of the sequence inside the domain, whereas diffuse FAPs characterizes how the mass concentrates on the boundary of the domain. Therefore, a diffuse FAP cannot be seen as the counterpart of an improper distribution. As an illustration, we show the difficulty to define a uniform FAP distribution on the natural numbers as an equivalent of the improper flat prior. We also show that expected logarithmic convergence may depend on the chosen sequence of compact sets.

Introduction

Improper priors and finitely additive probabilities (FAP) are the two main extensions of the standard Bayesian paradigm based on proper priors, i.e. countably additive probabilities (see Hartigan 1983, p.15). Both extensions induce paradoxical phenomena such as strong inconsistency [START_REF] Stone | Strong inconsistency from uniform priors[END_REF][START_REF] Dubins | Finitely additive conditional probabilities, conglomerability and disintegrations[END_REF] or marginalization paradoxes [START_REF] Dawid | Marginalization paradoxes in Bayesian and structural inference[END_REF]) that do not occur with proper priors. To have a better understanding of these phenomena, some authors such as [START_REF] Stone | Review and analysis of some inconsistencies related to improper priors and finite additivity[END_REF] or Kadane et al. (1986, p.218), consider improper distributions and FAPs as limits of proper prior sequences w.r.t. to appropriate topologies. Heuristically, this approach seems to establish a link between improper distributions and FAP.

Seeing a FAP as a limit is a way to preserve the total mass equal to 1, while sacrificing the countable additivity. This point of view has been mainly supported by [START_REF] De Finetti | Probability, induction and statistics. The art of guessing[END_REF]. On the other hand, improper distributions aim at preserving the countable additivity, while sacrificing a total mass equal to 1. Improper distributions appear naturally in the framework of conditional probability, see [START_REF] Rényi | On a new axiomatic theory of probability[END_REF] and more recently Taraldsen andLindqvist (2010, 2016) and [START_REF] Lindqvist | On the proper treatment of improper distributions[END_REF]. Conditional probability spaces are also related to projective spaces of measures [START_REF] Rényi | Foundations of probability[END_REF] which have a natural quotient space topology and a natural convergence mode, named q-vague convergence by [START_REF] Bioche | Approximation of improper priors[END_REF]. Bayesian inference with improper posterior is justified by [START_REF] Taraldsen | Statistics with improper posteriors[END_REF] from a theoretical point of view. [START_REF] Bord | A cautionary note on Bayesian estimation of population size by removal sampling with diffuse priors[END_REF] consider the convergence of proper distribution sequences to an improper posterior for Bayesian estimation of abundance by removal sampling. [START_REF] Tufto | Estimating brownian motion dispersal rate, longevity and population density from spatially explicit mark-recapture data on tropical butterflies[END_REF] propose to adapt MCMC for the estimation of improper posteriors.

In this paper, we mainly consider convergence of prior distributions to FAPs or to improper distributions, regardless to any statistical model. In Section 2, we define the notion of limits in the settings of improper distributions and of FAPs. We show that improper distributions and FAPs represent two distinct characteristics of a sequence of proper distributions. Therefore, they cannot be connected by the mean of proper distribution sequences. In Section 3, we revisit the notion of uniform distribution on integers in the light of our results. In Section 4, we illustrate with some examples the fundamental difference between convergence to an improper prior and to a FAP. In Section 5, we consider expected logarithmic convergence, defined by [START_REF] Berger | The formal definition of reference priors[END_REF] to approximate an improper distribution by a sequence of truncated proper priors. We apply some of the methods used in Section 2 to propose an example where this convergence mode depends on the chosen sequence of compact sets.

Convergence of probability sequences

We denote by C b the set of continuous real-valued bounded functions on a space Θ and by C K the set of continuous real-valued functions with compact support. For a σ-finite measure π, we denote π(f ) = f (θ) dπ(θ). Let {π n } n∈N be a sequence of proper distributions. The usual converge mode of {π n } n∈N to a proper prior π is the narrow convergence, also called weak convergence or convergence in law, defined by:

π n narrowly ------→ n→+∞ π ⇐⇒ π n (f ) -----→ n→+∞ π(f ) ∀f ∈ C b .
(1)

When it exists, the narrow limit of {π n } n is necessarily unique. In this section, we consider two alternative convergence modes when there is no narrow limit, and especially when the total mass tends to concentrate around the boundary on the domain, more precisely when lim n π n (f ) = 0 for all f in C K . The idea is to consider a proper prior either as a special case of FAP or as a special case of a Radon measure, and for each case, to define a convergence mode in a formalized way.

In the following, Θ is a locally compact separable metric space. This is the case, for example, for usual topological finite-dimensional vector spaces or for denumerable sets with the discrete topology. In the latter case, any function is continuous and a compact set is a finite set.

Convergence to an improper distribution

To extend the notion of narrow limits, we consider here proper distributions within the set of projective space of positive Radon measures as follows: we denote by R the set of non-null Radon measures, that is regular countably additive measures with finite mass on each compact set. Note that, in the discrete case, any σ-finite measure is a Radon measure.

We define an improper distribution as an unbounded Radon measure which appears in parametric Bayesian statistics (see, e.g. [START_REF] Jeffreys | Theory of probability[END_REF]). The projective space R associated to R is the quotient space for the equivalence relation ∼ defined by π 1 ∼ π 2 iff π 2 = α π 1 for some positive scalar factor α. To each Radon measure π is associated a unique equivalence class π = {π = α π ; α > 0}. Therefore, a projective space is a space where objects are defined up to a positive scalar factor. It is natural in Bayesian statistics to consider such projective space since two equivalent priors give the same posterior. The projective space R is also naturally linked with conditional probability spaces [START_REF] Rényi | On a new axiomatic theory of probability[END_REF]. All the results presented below on the convergence mode w.r.t. to the projective space R can be found in [START_REF] Bioche | Approximation of improper priors[END_REF]. The usual topology on R is the vague topology defined by

π n vaguely -----→ n→+∞ π ⇐⇒ π n (f ) -----→ n→+∞ π(f ) ∀f ∈ C K .
(2)

From the related quotient topology, we can derive a convergence mode, called q-vague convergence: a sequence {π n } n in R converges q -vaguely to a (non-null) improper distribution π in R if π n converges to π w.r.t. the quotient topology where π n and π are the equivalence classes associated to π n and π. The limit π is unique whereas π is unique only up to a positive scalar factor. It is not always tractable to check a convergence in the quotient space. However, there is an equivalent definition in the initial space R: {π n } n converges q -vaguely to π if there exists some scalar factors α n such that {α n π n } n converges vaguely to π:

π n q-vaguely -------→ n→+∞ π ⇐⇒ a n π n vaguely -----→ n→+∞
π for some a 1 , a 2 , ... > 0 .

(3)

The q-vague convergence can be considered as an extension of the narrow convergence in the sense that if {π n } n and π are proper distributions and {π n } n converges narrowly to π then {π n } n converges q-vaguely to π. Note that the converse part holds if and only if {π n } n is tight (see Bioche and Druilhet 2016, Proposition 2.8).

When a sequence {π n } n of proper distributions converges q-vaguely to an improper distribution, then lim n π n (K) = 0 for any compact K (Bioche and Druilhet 2016, Prop. 2.11). The following lemma gives an apparently stronger, but in fact equivalent, result. It will be useful to establish our main result and to construct examples in Section 4.3 and 5.

Lemma 1 Let {π n } n be a sequence of proper distributions such that lim n π n (K) = 0 for any compact K. Then there exists a non-decreasing sequence of compact sets K n such that ∪ n K n = Θ and lim n π n (K n ) = 0. Moreover, K n may be chosen such that, for any compact K, there exists an integer N such that K ⊂ K N .

Proof Let K m , m ≥ 1, be an increasing sequence of compact sets with ∪ m K m = Θ. For each m, lim n π n ( K m ) = 0, so there exists an integer N m such that N m > N m-1 and π n ( K m ) ≤ 1/m for n > N m . Consider now such a sequence of integers N m , m ≥ 1. For any n there exists a unique integer m such that N m ≤ n < N m+1 . We define

K n by K n = K m . So, π n (K n ) = π n ( K m ) ≤ 1/m.
Since m increases with n, lim n π n (K n ) = 0. Furthermore, the sequence K m can be chosen such that, for any compact K, K is a subset of all but finitely many K m , (see e.g. Bauer 2001, Lemma 29.8). By construction, the same property holds for the sequence K n .

Note that, lim n π n (K) = 0 for any compact set K does not imply that {π n } n converge q-vaguely, as shown in Section 4 on some examples.

Convergence to a FAP

Here, we consider proper distributions as special cases of FAPs. Denote by F b the set of bounded real-valued measurable functions on Θ. A FAP π is a linear functional on F b which is positive, i.e. π(f ) ≥ 0 if f ≥ 0, and which satisfies π(1) = 1. Therefore, the set of FAPs is included in the topological dual of F b equipped with the sup-norm. For any measurable set E ⊂ Θ, we define

π(E) = π(1 E ), where 1 E (x) = 1 if x ∈ E and 0 otherwise. We also denote f (θ) dπ(θ) = π(f ).
For most authors (see, e.g. [START_REF] Heath | On finitely additive priors, coherence, and extended admissibility[END_REF], a FAP is a linear functional on the set of bounded real valued functions. Here, we do impose a measurability condition, since we require proper distributions to be special cases of FAPs. In the case where Θ is a denumerable set equipped with the usual discrete topology, any function or set is measurable, and so, both definitions of FAPs are equivalent.

Let {π n } n be a sequence of FAPs. The usual convergence mode for FAPs is associated to the weak* topology: a sequence

{π n } n converges to π if lim n π n (f ) = π(f ) for any f ∈ F b .
When {π n } n does not converge, we may consider limit points, as proposed by [START_REF] Stone | Review and analysis of some inconsistencies related to improper priors and finite additivity[END_REF] for denumerable sets and extended here to more general sets. The existence of limit points relies on the Banach-Alaoglu-Bourbaki theorem (see, e.g. [START_REF] Rudin | Functional analysis[END_REF], since a FAP belongs to the unit ball in the dual of F b , which is compact for the weak * -topology. Hence, for any sequence {π n } n of FAPs, there exists at least one limit point π which is defined as a FAP limit. We recall that π is a limit point of {π n } n for the weak * -topology if and only if for any integer p, any f 1 , ...f p in F b and any ε > 0, there exists an infinite number of n such that

|π n (f i ) -π(f i )| ≤ ε, i = 1, ..., p.
Since F b is not in general first-countable, there does not necessarily exist a subsequence {π n k } k that converges to π. We can only say that, for any f 1 , ..., f p in F b , there exists a subsequence

{π n k } k such that (π n k (f 1 ), ..., π n k (f p )) converges to (π(f 1 ), ..., π(f p )).
When π n and π are proper distributions, then {π n } n converging narrowly to π does not imply that π is a FAP limit point of {π n } n . Therefore, unlike q-vague limits, FAP limit points cannot be considered as an extension of the narrow convergence. For example, consider the proper distributions π n = δ √ 2/n , where δ is the Dirac measure. The sequence {π n } n converges narrowly to π = δ 0 but π is not a FAP limit point of {π n } n . To show this, consider f (θ) = 1 Q (θ) ∈ F b , with Q the set of rational numbers, we have lim n π n (f ) = 0 = π(f ) = 1. We can only say that any FAP limit point of the sequence {π n } n will coincide with π = δ 0 on the set C b .

To consider a FAP limit as an extension of the narrow convergence, we should have defined FAPs on the space C b rather than F b . However, with this choice, π(E) is not well defined for all measurable sets E.

In the special case where Θ is a denumerable set, any real-valued function on Θ is continuous. So, if a sequence of proper distributions {π n } n converges narrowly to a proper distribution π, then π is a FAP limit point.

Another way to extend the notion of limit can be obtained by using the Hahn-Banach theorem as follows (see [START_REF] Huisman | Infinitesimal distributions, improper priors and Bayesian inference[END_REF]

: let S c be the set of f ∈ F b such that lim n π n (f ) exists. A FAP π is said to be an extended FAP limit of {π n } n if lim n π n (f ) = π(f ) for any f ∈ S c , and if π(f ) ≤ lim sup n π n (f ).
The existence of a FAP π satisfying this requirement is guaranteed by the Hahn-Banach theorem (see [START_REF] Rudin | Functional analysis[END_REF]: define the linear function Φ on S c by Φ(f ) = lim n π n (f ) and the sublinear functional p(f ) = lim sup n π n (f ). Then, there exists a linear functional π on F b that coincides with Φ on S c and that satisfies π(f ) ≤ p(f ) on F b . The condition π(f ) ≤ p(f ) implies that π is a FAP. Conversely, an extended FAP limit necessarily satisfies π(f ) ≤ p(f ). Replacing f by -f gives π(f ) ≥ lim inf n π n (f ). Therefore, an extended FAP limit can be characterized by the following lemma:

Lemma 2 A FAP π is an extended FAP-limit of the sequence {π n } n if and only if for any

f ∈ F b lim inf n π n (f ) ≤ π(f ) ≤ lim sup n π n (f ) (4)
or equivalently if and only if for any measurable set E

lim inf n π n (E) ≤ π(E) ≤ lim sup n π n (E). ( 5
)
Note that the sequence {π n } n converges to π for the weak * topology if and only if S c = F b . In general, an extended FAP limit is not unique and its existence relies on the axiom of choice.

The set of limit points of {π n } n is included in the set of extended FAP limits. The converse inclusion is false in general. Inequalities (4) or ( 5) hold for limit points but are not sufficient to characterize them. It is easy to see that the closed convex hull of the set of limit points is included in the set of extended FAP limits. We conjecture that, conversely, the set of extended FAP limits defined by ( 4) is the closed convex hull of the set of limit points. As a simple example, consider the sequence {π n } n with π 2n = δ 0 and π 2n+1 = δ 1 . There are only two limit points δ 0 and δ 1 , whereas any π = αδ 0 + (1 -α)δ 1 , 0 ≤ α ≤ 1 is a extended FAP limit. In Section 4.1, we illustrate the difference between these two constructions of limits with another example.

Even if the notion of FAP limit points is more restrictive than the notion of extended FAP limit, the main results, especially Theorem 1, Corollary 1, Proposition 2, Lemma 3 and 4 hold for both of them. In the following, we consider only FAP limit points.

FAP limit points vs q-vague convergence

The fact that a sequence of proper distributions has both improper and FAP limit points may suggest a connection between the two notions as proposed heuristically by several authors, such as [START_REF] Levi | The Enterprise of Knowledge The[END_REF]; [START_REF] Stone | Review and analysis of some inconsistencies related to improper priors and finite additivity[END_REF]; [START_REF] Kadane | Statistical implications of finitely additive probability[END_REF]. The following results show that this is not the case. Roughly speaking, it is shown that any FAP which is a limit point of some proper distribution sequence can be connected to any improper prior by this mean.

Theorem 1 Let {π n } n be a sequence of proper distributions such that lim n π n (K) = 0 for any compact set K. Then, for any improper distribution π, it can be constructed a sequence { π n } n which converges q-vaguely to π and which has the same set of FAP limit points as {π n } n .

Proof For any FAP or any proper or improper distribution µ, we define the distribution (1 A µ) by (1 A µ)(f ) = µ(1 A f ) where A is any measurable set.

From Lemma 1, it can be constructed an exhaustive increasing sequence K n of compact sets such that lim n π n (K n ) = 0. Put γ n = π n (K n ) and define the sequence of proper distributions

π n = γ n 1 π(Kn) 1 Kn π + (1 -γ n ) 1 πn(K c n ) 1 K c
n π n , with K c the complement of K. By Lemma 4 and 3 in Appendix A, π n has the same FAP limit points as {π n }. By Lemma 5, π n converges q-vaguely to π.

Corollary 1 Let {π n } n be a sequence of proper distributions that converges q-vaguely to an improper distribution π (1) . Then, for any other improper distribution π (2) , it can be constructed a sequence { π n } n that converges q-vaguely to π (2) and that has the same FAP limit points as {π n } n .

We have shown that no direct link between an improper limit and a FAP limit point can be established. One can only say that, if a sequence of proper distributions converges to an improper distribution, its FAP limit point π are diffuse, i.e. π(K) = 0 for any compact set K.

Uniform distribution on integers

In this section, we compare different notions of uniform distributions on the set N of integers, by using several considerations such as limit of proper uniform distributions.

We also illustrate the fact that FAP uniform distributions are not well defined objects (de Finetti 1972, pp.122,224). Contrary to uniform improper distributions, FAP limit points of uniform distributions on an exhaustive sequence of compact sets are highly dependent on the choice of that sequence.

Uniform improper distribution

There are several equivalent ways to define a uniform improper prior on integers. These definitions lead to a unique, up to a scalar factor, distribution. The uniform distribution can be defined directly as a flat distribution, i.e. π(k) ∝ 1 for any integer k. It is the unique (up to a scalar factor) measure that is shift invariant, i.e. such that π(k + A) = π(A) for any integer k and any set of integers A. The uniform distribution is also the q-vague limit of the sequence of uniform proper distributions on K n = {0, 1, ..., n}. More generally and equivalently, the uniform distribution is the q-vague limit of any sequence of proper uniform priors on an exhaustive increasing sequence {K n } n of finite subsets of integers.

Uniform finitely additive probability

The notion of uniform finitely additive probabilities is more complex. Contrary to the improper case, there is no explicit definition since π(k) = 0 for any integer k. We present here several non equivalent approaches to define a uniform FAP. The first two ones can be found in [START_REF] Kadane | Using finitely additive probability: uniform distributions on the natural numbers[END_REF] and [START_REF] Schirokauer | Uniform distributions on the natural numbers[END_REF].

Shift invariant (SI) uniform distribution

As for the improper case, a uniform FAP π can be defined as being any shift invariant FAP, i.e. a FAP satisfying π(A) = π(A + k) for any subset of integers A and any integer k. Such a distribution will be called SI-uniform. In that case, one necessarily has :

π(k 1 + k 2 × N) = k -1 2 , for any (k 1 , k 2 ) ∈ N × N * .
In [START_REF] Kadane | Using finitely additive probability: uniform distributions on the natural numbers[END_REF], the authors investigate the properties of FAPs satisfying only π

(k 1 + k 2 × N) = k -1
2 , where the sets k 1 + k 2 × N are called residue classes.

Limiting relative frequency (LRF) uniform distributions.

Kadane and O'Hagan (1995) consider a stronger condition to define uniformity. For a subset A, define its Limiting Relative Frequency LRF(A) by

LRF(A) = lim N →∞ #{k ≤ N, s.t. k ∈ A} N + 1 ,
when this limit exists. A FAP π on N is said to be LRF uniform if π(A) = p when LRF(A) = p. Let π n be the uniform proper distribution on K n = {0, 1, ..., n}, then LRF(A) = lim n→∞ π n (A). Therefore, any FAP limit point of π n is LRF uniform. In fact, a FAP π is LRF uniform if and only if it is an extended FAP limit of {π n } n .

It is worth noting that, unlike the q-vague limit, FAP limit points are highly dependent on the choice of the increasing exhaustive sequence of finite sets K n . Changing the sequence {K n } n changes the notion of uniformity. For example, if π n is the uniform distribution on

K n = {2k ; 0 ≤ k ≤ n 2 } ∪ {2k + 1 ; 0 ≤ k ≤ n}, then lim n π n (2N) = 1, whereas lim n π n (2N) = 1/2.

Bernoulli Scheme (BS) uniform distribution

We propose here another notion of uniformity that is not dependent of the choice a particular increasing sequence of finite sets K n as for the LRF uniformity. Consider a Bernoulli Scheme, that is, a sequence {X k } k∈N of i.i.d. Bernoulli distributed random variables with mean p ∈ [0, 1]. Define the random set A(X) = {k ∈ N, s.t. X k = 1}. A FAP π is said to be BS-uniform if, for any p ∈ [0; 1], π(A(X)) = p, almost surely. By the strong law of large numbers, LRF(A(X)) = p, almost surely.

Proposition 1 Let {K n } be an increasing sequence of finite subsets of N, with ∪ n∈N K n being infinite. Then any FAP which is a limit point of the sequence π n of uniform distributions on K n is BS-uniform.

When ∪ n∈N K n = N, this proposition shows that any FAP limit point of uniform distribution sequences is BS-uniform. In particular, a LRF uniform FAP is also BS uniform. However, if for example K n is the set of even numbers less or equal to n, then any FAP limit point π of the sequence of uniform distributions on K n is BS-uniform. However, π is not uniform on N but uniform on 2N. Therefore, BS uniformity looks much more like a necessary condition for a FAP to be uniform, than like a complete definition.

Comparison of convergence modes on examples

We consider here some examples that illustrate the difference between convergence of proper distributions to an improper distribution and to a FAP.

FAP limit points on N.

For a sequence {π n } n of proper distributions on N, it is known that there does not necessarily exist a q-vague limit, but if it exists, it is unique up to a scalar factor, i.e. it is unique in the projective space of Radon measures. At the opposite, we have seen that a FAP limit point always exists but is not necessarily unique.

We illustrate the non-uniqueness of FAP limit point with an extreme case. Consider the sequence of proper distributions π n = δ n , where δ n is the Dirac measure on n. This sequence has no q-vague limit since π n (k) = 0 for n > k.

Let π be a FAP limit points. For any subset A, there exists a subsequence {π n k } such that π n k (A) convergences to π(A). So, π(A) ∈ {0, 1}. Therefore π is any remote FAP, that is a diffuse FAP such that π(A) ∈ {0, 1}, as defined by Dubins (1975, p. 92). This also proves the existence of remote FAPs. Note that a remote FAP is neither BS uniform nor SI and therefore cannot be LRF uniform. As a remark, the extended FAP limits of π n are all the diffuse FAPs.

Convergence of sequence of Poisson distributions

We consider here the sequence {π n } n of Poisson distributions with mean n. Although lim n π n (K) = 0 for any finite set K, this sequence of proper distributions does not converge q-vaguely to any improper distribution (Bioche and Druilhet 2016, § 5.2). As a remark, let π n be the shifted measures defined on the set of positive and integers integers Z by π n (B) = π n (B +n), where π n can be seen as a measure on the set Z, with π n (k) = 0 for k < 0. Then, using the approximation of the Poisson distribution by an normal distribution, it can be shown that the sequence π n converges q-vaguely to the improper uniform measure on the set Z.

We consider now the FAP limit point of the sequence {π n } n . The next result shows that these limits have some properties of uniformity described in Section 3 but not all of them. The proof is given in Appendix B.

Proposition 2 Any FAP π which is a limit point of the sequence {π n } n of Poisson distribution with mean n is SI-uniform and BS-uniform but not necessarily LRF-uniform.

Therefore, the FAP limit points of the Poisson distribution sequence are examples of SI-and BS-uniform distributions that are not LRF uniform. [START_REF] Kadane | Uniform distributions on the integers: a connection to the Bernouilli random walk[END_REF] give another example of SI but not LRF uniform FAPs using paths of random walks. Even if they consider FAPs on a subset of bounded functions, it can be extended to F b by using the Hahn-Banach theorem similarly to Section 2.2.

FAP vs q-vague convergence of uniform proper distributions

To illustrate the fact that any FAP limit point can be related with any improper distribution, consider again the sequence {π n } n of Poisson distributions with mean n and let π 0 be any improper distribution on the integers. Since lim n π n (K) = 0 for any finite set, from Lemma 1 we can construct an exhaustive sequence of finite set K n such that lim n π n (K n ) = 0. Put K n = {k ∈ N, k ≤ n/2}, which satisfies this condition. Define the sequence of proper distributions π n by :

π n (A) = π n (K n ) π 0 (A ∩ K n ) π 0 (K n ) + (1 -π n (K n )) π n (A ∩ K c n ) π n (K c n ) (6)
for any set A. From Theorem 1, { π n } n converges q-vaguely to π 0 and has the same FAP limit points as {π n } n .

As another example, let {π n } n be the sequence of uniform distributions on {0, 1, ..., n} and choose

K n = {k ∈ N, k ≤ √ n}.
Then, lim n π n (K n ) = 0. Therefore, for any improper distribution π 0 on the set of integers, the sequence constructed as in (6) has the same FAP limit points as those of the sequence of uniform distributions {π n } n and converges q-vaguely to π 0 . This shows again the difficulty to connect the uniform improper distribution and uniform FAPs by limits of proper distributions.

Convergence of beta distributions

In this section, we consider the limit of the sequence of Beta distribution π n = Beta(a n , b n ) defined on Θ =]0, 1[ when a n and b n go to 0. We will see that, contrary to the improper limit, the FAP limit points depend on the way a n and b n go to 0. This illustrates again the difference between the two kinds of limits.

The density of a beta distribution Beta(a, b) is given by

π a,b (x) = 1 β(a, b) x a-1 (1 -x) b-1 for x ∈]0; 1[
where β(a, b) is the beta function.

From [START_REF] Bioche | Approximation of improper priors[END_REF], the unique (up to a scalar factor) qvague limit of Beta(a n , b n ) when a n and b n go to 0 is the Haldane improper distribution:

π H (x) = 1 x(1 -x)
for x ∈]0; 1[. The q-vague limit gives no information on the relative concentration of the mass around 0 and 1: for 0

< u < v < 1, π H (]0, u]) = π H (]v, 1[]) = +∞.
To explore this concentration, we temporarily replace the space Θ by Θ = [0, 1]. This has no consequence on the Beta distributions but the Haldane distribution is no longer a q-vague limit of the sequence. Put c n = a n /b n and assume that {c n } n converges to some c ∈ [0, 1]. The sequence {π n } n converges narrowly, and hence q-vaguely, to the proper distribution π = 1 1+c δ 0 + c 1+c δ 1 . Contrary to the Haldane prior, π shows how the mass concentrates on the boundary of the domain, but gives no information on the behavior of the sequence inside the domain. Note that the Haldane distribution is not a Radon measure on Θ since π H ([0, 1]) = +∞ where [0, 1] is a compact set. Therefore π H cannot be a candidate for the q-vague limit on Θ.

We now consider the FAP limit points on Θ =]0, 1[ of π n and we show that they give an information similar to that given by π on the way the mass concentrate on the boundary of the domain. Again, we assume that c n = a n /b n converges to some c ∈ [0, 1]. Easy calculations show that for any 0 < ε < 1 lim n π n (]0, ε[) = 1 1+c and lim n π n (]1-ε, 1[) = c 1+c . Therefore, for any FAP limit point π and for any ε ∈]0, 1[, we have π(]0, ε

[) = 1 1+c and π(]1 -ε, 1[) = c 1+c , with π([u, v]) = 0 for 0 < u < v < 1.

Expected logarithmic convergence

In Bayesian statistics, consider a statistical model p(x|θ) and an improper prior π(θ) on Θ. Define the truncated proper prior π n (θ) ∝ π(θ) 1 θ∈Kn , for some exhaustive increasing sequence of compact sets {K n } n . From [START_REF] Berger | The formal definition of reference priors[END_REF], a sequence of posteriors distributions π n (θ|x) is said to be expected logarithmically convergent to π

(θ|x) if lim i→∞ X p i (x)κ(π(•|x), π i (•|x))dx = 0 ,
where p i (x) = Θ p(x|θ)π i (θ)dθ, and κ(m 1 , m 2 ) denotes the Kullback-Leibler distance between probability measures m 1 and m 2 . The prior π is said to be permissible w.r.t. p(x|θ) if π(θ|x) is proper and if there exists some exhaustive sequence of compact sets {K n } n such that π n (θ|x) is expected logarithmically convergent to π(θ|x). Note that π n (θ|x) converges q-vaguely to π(θ|x), provided that p(x|θ) is continuous w.r.t. θ for any x (Bioche and Druilhet 2016, Prop. 3.1).

An open problem is to know whether this property is always independent from the choice of the sequence of compact sets {K n } n . We present here a situation where it is not.

The construction of this counter-example relies on the fact that the tail behavior of a sequence of distributions is not directly related to its q-vague convergence as explained in Section 2.3.

Consider the following model: for any integers x and θ (included negative integers) define:

p(x|θ) =    1 3 if θ ≥ 1, x ∈ { θ 2 , 2θ, 2θ + 1} 1 if θ ≤ 0, x = θ 0 otherwise
where [l] the integer part of l, with the particular case [1/2] = 1. For θ ≤ 0, x ≤ 0, we have a deterministic model. Remark also that, for x, θ ≥ 1, we have the equivalence :

x ∈ θ 2 , 2θ, 2θ + 1 ⇔ θ ∈ x 2 , 2x, 2x + 1 . Consider the flat prior π(θ) ∝ 1. If we consider only x, θ ≥ 1, this model corresponds to a model proposed by Fraser Monette and Ng (1985) and used by [START_REF] Berger | The formal definition of reference priors[END_REF] to illustrate their approach.

Let K n = {-a n ≤ θ ≤ b n } be an exhaustive sequence of compact sets, with a n , b n → +∞. Denote I n = X p n (x)κ(π(•|x), π n (•|x))dx. We have:

I n = 1 a n + b n + 1 ln(3) 2b n + 1 -bn 2 3 + ln(3/2) 3 ω(b n )
where ω(b n ) = 1 if b n is even, and 0 if b n is odd. Therefore, as a n and b n tend to infinity, I n ∼ ln(3) 2 bn an+bn+1 . When b n /a n tends to 0, lim n→∞ I n = 0, which gives an expected logarithmically convergent sequence of posteriors. However, taking a n = b n leads to lim n→∞ I n = ln(3) 4 , and the sequence of corresponding posteriors is not expected logarithmically convergent.

This example shows that, at least for some statistical models p(x|θ) and improper prior π(θ), the notion of expected logarithmic convergence may depend on the choice of the sequence of compact sets. It could be interesting to characterizes situations where it does not. This is left for future works.

Conclusion and perspectives

In this paper, we have shown that the characteristics of a sequence of proper distributions given by its FAP or improper limits are quite different. As a consequence, there is no clear link between improper distributions and FAPs: a diffuse FAP cannot be considered as the counterpart of some improper distribution.

In Bayesian statistics, improper distributions are commonly used in practice, even if some paradoxes may occur. They are easy to interpret either through their densities or as conditional probabilities [START_REF] Taraldsen | Conditional probability and improper priors[END_REF]. At the opposite, diffuse FAPs are never used in practice, mainly because their constructions are always implicit and because diffuse FAPs give information only on the boundary of the parameter space, which is difficult to construe.

However, FAPs may provide a better understanding of some limit behavior that are not captured by improper distributions. The fact that our main results rely on explicit constructions of proper prior sequences may be useful to provide counterexamples. For example, in Section 5, we have shown that the notion of expected logarithmic convergence may depend on the sequence of compact sets. We hope to use our results in future works to have a better understanding of some paradoxical phenomena in Bayesian statistics, such as strong inconsistency or the marginalization paradox.

Proof First note that, for any set A , LRF (A ) = p if, and only if, {k ≤ n, k ∈ A } = pn + o(n). Therefore, for any set A with LRF (A) = p and for any set B such that {k ≤ n, k ∈ B} = o(N ), one has both LRF (A ∪ B) = p and LRF (A \ B) = p. Take now for set B the following :

B = k∈N u ∈ N : 4 k -2 k k ≤ u ≤ 4 k + 2 k k .
For that B, one has :

lim sup n→∞ {k ≤ n, k ∈ B} n + 1 = lim k→∞ k i=0 2 i+1 i 4 k + 2 k k ≤ lim k→∞ (k + 1)2 k+2 4 k = 0,
and thus LRF (B) = 0. However, π 4 k (B) converges to 1. Indeed, if U k is some random variable with law π 4 k , one has :

π 4 k u ∈ N : 4 k -2 k k ≤ u ≤ 4 k + 2 k k = P U k -4 k √ 4 k ∈ -k ; k .
The right-hand side term above converges to 1 thanks to the central limit

  theorem. Hence LRF (A ∪ B) = LRF (A \ B) = p while π 4 k (A ∪ B) converges to 1, and π 4 k (A \ B) converges to 0. Now, for any p ∈ [0; 1], choose two numbers a < b, so that p = b a e -u 2 /2 √ 2π du. Take the set B to be :B = k∈N u ∈ N : 4 k + 2 k max(-k, a) ≤ u ≤ 4 k + 2 k min(k, b) ,then LRF (B ) = 0 again and π 4 k (B ) converges to p , still thanks to the central limit theorem. Let A = (A \B)∪B . Then LRF (A) = p and lim k→∞ π 4 k (A) = p . Now, any FAP limit point π of subsequence {π 4 k } k is also a FAP limit point of {π k } k . Hence, π is SI-uniform and BS-uniform, but one has π(A) = p and LRF (A) = p.

Appendix A

We establish some lemmas useful to prove Theorem 1. The first one is straightforward.

Lemma 3 Let {π

(1) n } n and {π

(2) n } n be two sequences of proper distributions and 0 ≤ γ n ≤ 1 be a sequence of scalars that converges to 0. Then, the sequence defined by π n = γ n π

(2) n has the same FAP limit points as {π

Lemma 4 Let {π n } n be a sequence of proper priors and K n be a non-decreasing sequence of compact sets such that lim n π n (K n ) = 0, then the sequence defined by

n π n has the same FAP limit points as {π n } n .

Proof First, note that {π n } n is not defined when π n (K n ) = 1, but this cannot occur more than a finite number of times. For any

At the opposite of Lemma 4, the following lemma shows that if we consider the restriction of a sequence {π n } n of a proper or improper distribution on a exhaustive increasing sequence {K n } n of compact sets, we preserve the q-vague limits.

Lemma 5 Let K n be a non-decreasing sequence of compact sets such that ∪ n K n = Θ and such that, for any compact K, there exists N such that K ⊂ K N . A sequence {π n } n of Radon measures converges q-vaguely to the Radon measure π if and only if π n = 1 πn(Kn) 1 Kn π n converges q-vaguely to π. Proof Assume that π n converges q-vaguely to π, then there exists some positive scalars {a n } n such that for any f in C K , lim n a n π n (f ) = π(f ). Put a n = a n π n (K n ) and denote by K f a compact set that includes the support of f . Then, there exists an integer N such that K f ⊂ K n for n > N . Therefore, for n > N , a n π n (f ) = a n π n (f ). The result and its reciprocal follow.

Appendix B

We prove here Proposition 2 of section 4.2.

In order to show that π is SI-uniform, we consider π n as a distribution on the set of positive and negative integers, extending it by 0 on the non-positive integers. Define by π

, where • T V is the total variation norm. Therefore, for any subset of N, lim n→∞ |π n (A + k) -π n (A)| = 0. Letting n go to infinity, we deduce that, for any FAP limit point π of π n , and any integer k : π(A + k) = π(A).

The fact that π is BS-uniform comes from an easy adaptation of the Hoeffding inequality in that context. Let (X k ) k∈N be a Bernoulli scheme, of parameter p, and denote by P the associated probability. Hoeffding inequality gives, that, for any n :

, for some positive constant c. The expected conclusion is then obtained thanks to the Borel-Cantelli lemma.

The fact that some of the limit points π of {π n } n are not LRF uniform is a direct consequence of the following lemma.

Lemma 6 For any 0 ≤ p, p ≤ 1, there exists a set A and some FAP limit points π of {π n } n such that LRF (A) = p and π(A) = p .


Kévin Billon

Morvan Ouisse
email: morvan.ouisse@femto-st.fr

Manuel Collet

Emeline Sadoulet

E Sadoulet-Reboul

Design of smart metamaterials for vibration control: extension of Bloch approach to handle finite system boundary conditions

Keywords: Periodic structures, Metamaterial, Dispersion, Dissipation, Vibroacoutic

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

A periodic medium is a material or a structural system that exhibits spatial periodicity. [START_REF] Brillouin | Wave propagation in periodic structures[END_REF] The study of periodic structures has a long history in the field of vibrations and acoustics. [START_REF] Hussein | Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook[END_REF] This topic has interested researchers over the years, and a growing activity on this field is observed on the last years, with the objective of designing structures exhibiting properties that conventional ones cannot possess. [START_REF] Shan | Design of planar isotropic negative poissons ratio structures[END_REF][START_REF] Slann | Cellular plates with auxetic rectangular perforations[END_REF][START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF] The methods currently used are most of the time based on those derived from wave propagation in crystals, [START_REF] Joannopoulos | Photonic crystals: molding the flow of light[END_REF] where almost no dissipation occurs. The classical Floquet-Bloch approach 7, 8 is a method commonly used for the study of periodic structures. The material constitutive law is linear, elastic and isotropic. The periodicity is defined on the borders of the domain u R = e -jkxr u L and v R = e -jkyr v L where u R (resp. v R ) is the displacement on the right border and u L (resp. v L ) is the displacement on the left border in x (resp. y) axis, k x and k y are respectively the wavenumbers in the x and y directions 9, 10 for details.

METAMATERIAL DESIGN

The metamaterial consists in an infinite periodic bidirectional waveguide. [START_REF] Wu | Waveguiding and frequency selection of lamb waves in a plate with a periodic stubbed surface[END_REF][START_REF] Billon | A two-state phononic crystal using highly dissipative polymeric material interface[END_REF][START_REF] Billon | Design and experimental validation of an adaptive phononic crystal using highly dissipative polymeric material interface[END_REF] It is a 1 mm thick plate with periodic cylindrical pillars. In order to design a two-state phononic crystal, a polymeric base is included in the pillar (figure 1). The base plate is made of isotropic Aluminium 6063-T83 (ν = 0, 33, E = 69 GP a and ρ = 2700 kg/m 3 ). Pillars are made of combination between a highly dissipative polymer tBA/PEGDMA (ν poly = 0, 37, E poly , defined in the following and ρ poly = 1004[kg/m 3 ]) [START_REF] Butaud | Contribution à l'utilisation des polymères à mémoire de forme pour les structures à amortissement contrôlé[END_REF] and Aluminum 6063-T83. The polymer thickness has been chosen in order to open a resonant bandgap under the bragg bandgap frequency. According to the manufacturing process, the maximum thickness obtained after the UV photopolymerization is 3 mm. The natural frequency is around 40 kHz in this case. The eigenshape of the corresponding mode is shown in figure 2. A suitable model is required for the description of the frequency-dependent behaviour of the polymer. In the harmonic regime, the Young's modulus of the polymer is complex, dependent on frequency and temperature,

Polymer

E * poly (ω) = E + jE = E × (1 + j × tan(δ)), (1)
where tan(δ) = E /E is the loss factor, E is the storage modulus and E is the loss modulus. In this work, a fractional derivative Zener 15 model is used. This material exhibits a strong temperature dependency but in this case, at ambient temperature 25 • C, E * poly is constant (E = 2.21 GP a and tan(δ) ≈ 0).

DYNAMICAL PROPERTIES

Dispersion analysis

The "Shifted-cell operator" technique is used to obtain dispersion curves along the Γ -X direction in. [START_REF] Collet | Floquet-bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF] Damping is included in the analysis according to complex polymer Young's modulus using fractional derivative Zener model and aluminum loss factor equal to 0.5%.

Figure 3a shows dispersion curves along the Γ-X direction obtained with the "Shifted-cell operator" method using branch tracking. The associated group velocity is plotted figure 3b. The bandgap is visible around the selected frequency (40 kHz) linked to the design of the resonator.

Finite structure

The main goal of this part is to validate in a finite structure the phenomenon observed on an infinite structure.

A finite element model is presented. This is followed by an experimental validation. The metamaterial (21 × 7 × 0.1 cm 3 ) includes an interface composed by 7 × 7 unit cells (figure 4). A piezoelectric patch with harmonic voltage (|U | = 100 V ) is included in the model in order to be close to the experimental set up and cover all the frequency range from 0 to 50 kHz. Energy can propagate through the lattice at 34.4 kHz (figure 6b) and 50.0 kHz (figure 6e), these 2 frequencies are outside the predicted bandgap. At 39.2 kHz, inside the bandgap, the attenuation is widely visible, all the energy is confined in the input plate (IN).

At 43.3 kHz, the lattice is not efficient despite the fact that the frequency of interest is located inside the predicted bandgap. After shapes analysis, the vibration level is near to in the middle of the lattice but the energy propagates along the edges of the periodic lattice. Energy transmission occurs along the edges, and the dispersion analysis can not predict this effect.

Experimental validation

The metamaterial is made of metal and polymer. Polymer parts are realised by laser cutting and metallic part by classical mechanical processing (figure 7). The metamaterial is assembled in two steps by bonding parts together.

Polymer cylinders are sticked on the support plate. After drying and load process, at ambient temperature during 24 hours, aluminum cylinders are sticked on polymer cylinders. Piezoelectric patch provides harmonic excitation up to 50 kHz.

The metamaterial is suspended to reproduce free boundary condition as shown in figure 8.

The white noise generator provides random harmonic voltage (5 V ) between 500Hz and 50kHz, this signal is amplified 20 times by the piezoelectric amplifier. A 3D vibrometer is used to measure the velocity and frequency responses are calculated with H1 estimator.

The experimental analysis is focused on the 4 frequency points pointed out in the numerical part. Figures 9 show experimental deformed shapes at 33.4 kHz, 38.2 kHz, 42.2 kHz and 50.0 kHz corresponding to numerical deformed shapes (Figure 6). Shapes are similar between numerical and experimental results. In particular, it can clearly be observed in figure 9c that the energy is transported along the borders. This observation being experimentally validated, the next section aims at proposing a new methodology to identify configurations using cell-based computation.

. EXTENSION OF BLOCH APPROACH

Two connected cells are considered in the analysis (figure 10). Bondary conditions are as follow:

• in the y direction, classical Floquet-Bloch conditions are used, namely v R = e -jkyr v L where v R is the displacement on the right border and v L is the displacement on the left border in y axis,

• the x direction, displacement on the left border is linked to the displacement of the connected line u R = e +jkxr u L , while the right border uses the effective boundary conditions (free end in our case).

v L v R u L u R Free Figure 10
. Boundary conditions of the 2 cells system.

11a shows dispersion curves along Brillouin zone obtained with the "Shifted-cell operator" method for the 1 cell and 2 cells system. The blue curve corresponds to the 1 cell dispersion curve (figure 3a). The dispersion for the 2 cells system is plotted in red, a branch appears in the bandgap and the corresponding deformed shape is visible in the figure 11b. The shape is the same as the eigenshape but energy can propagate in the y direction along the free border.

CONCLUSION

In this article, a metamaterial mixing aluminum with a dissipative polymer interface is designed. A 3D finite element model of the metamaterial elementary cell is designed for dispersion analysis using the "Shifted-cell operator" method. In order to validate the design of the metamaterial, a complete 3D model integrating an interface composed of a distributed set of unit cells is presented. The attenuation in finite structure is not as important as that predicted by the bandgap in infinite structure. Energy propagates along each edges of the periodic lattice. This point is confirmed through experimental measurements. Boundary conditions are really important in this case and the dispersion analysis can not predict this effect. Finally, an extension of the Bloch approach is proposed to handle finite system boundary conditions in order to identify situations where energy transfer may arise because of reflections on the border of the domain.

Figure 1 .

 1 Figure 1. Metamaterial with 3 mm height polymeric interface between base and 7 mm height aluminum cylinder.

Figure 2 .

 2 Figure 2. Eigenshape at 40 kHz.

Figure 3

 3 Figure 3. a) Dispersion curves along the Γ -X direction obtained with the "Shifted-cell operator" method. b) Group velocity associated to dispersion curves. Grey shapes represent bandgaps.

Figure 4 .

 4 Figure 4. Finite structure with an interface composed by 7 × 7 unit cells distributed. Piezoelectric patch (PZT) for excitation in the bottom left corner.

Figure 5 .

 5 Figure 5. Numerical frequency responses. Average squared velocity amplitude |V z| 2 for the input plate (IN) and the output plate (OUT) respectively in blue and red. The grey shape represents the bandgap predicted by the dispersion analysis.

Figure 6a represents numerical

  Figure 6a represents numerical frequency responses focus on the frequency range of the bandgap. 4 points are selected in the frequency response, 2 outside (34.4 kHz, 50.0 kHz) and 2 inside (39.2 kHz, 43.3 kHz) the predicted bandgap. Corresponding operational deflections are visible on figures 6b, 6c, 6d and 6e.

  Figure 6. a) Zoom on numerical frequency responses. Average squared velocity amplitude |V z| 2 for the input plate (IN) and the output plate (OUT) respectively in blue and red. The grey shape represents the bandgap predicted by the dispersion analysis. Several points are selected with corresponding deformed shapes b) at 34.4 kHz, c) 39.2 kHz, d) 43.3 and e) 50.0 kHz.

Figure 7 .

 7 Figure 7. Metamaterial after several manufacturing steps.

Figure 8 .

 8 Figure 8. Metamaterial in suspended configuration.

Figure 9 .

 9 Figure 9. Experimental deformed shapes a) at 33.4 kHz, b) 38.2 kHz, c) 42.2 kHz and d) 50.0 kHz corresponding to numerical deformed shapes (Figures 6).

4

 4

Figure 11

 11 Figure 11. a) Dispersion curves along the Brillouin zone obtained with the "Shifted-cell operator" method for the 1 cell and 2 cells system. b) Deformed shape at 41 kHz.

ACKNOWLEDGMENTS

This work was financed by The French National Research Agency under grant number ANR-12-JS09-008-COVIA. It has been performed in cooperation with the Labex ACTION program (ANR-11-LABX-0001-01).