Simon Petitrenaud
email: simon.petit-renaud@univ-lemans.fr

Vincent Jousse
email: vincent.jousse@univ-lemans.fr

Sylvain Meignier
email: sylvain.meignier@univ-lemans.fr

Yannick Estève
email: yannick.esteve@univ-lemans.fr

Automatic named identification of speakers using belief functions

Keywords: Speaker identification, speaker recognition, information fusion, belief functions

In this paper, we consider the extraction of speaker identity (first name and last name) from audio records of broadcast news. Using an automatic speech recognition system, we present improvements for a method which allows to extract speaker identities from automatic transcripts and to assign them to speaker turns. The detected full names are chosen as potential candidates for these assignments. All this information, which is often contradictory, is described and combined in the Belief Functions formalism, which makes the knowledge representation of the problem coherent. The Belief Function theory has proven to be very suitable and adapted for the management of uncertainties concerning the speaker identity. Experiments are carried out on French broadcast news records from a French evaluation campaign of automatic speech recognition.

Introduction

In order to allow later retrieval of recorded information, large collections of audio documents have to be indexed. The system presented in this paper focuses on speaker identification by their full name in audio documents. The speaker identity detection is composed of several steps and is in most cases subject to uncertainty and confusion. The first step to automatically get audio documents indexing consists in detecting speakers turns and regrouping those uttered by the same speaker. It is generally based on a first stage of segmentation that consists in partitioning the regions of speech into homogeneous audio segments which contains ideally the voice of only one speaker, followed by a clustering stage that consists in giving the same label to segments uttered by the same speaker. A speaker turn starts when a speaker is starting to speak and ends when another speaker is starting to speak, or a song or advertising is starting. Speaker turns are regrouped by class of same but anonymous speakers. The next step is to automatically transcribe the content of speaker turns into words and is complemented by an annotation for some words as "named entities" or categories. Some words are particularly identified as "PERSON". The more promising way to identify speakers by their real full name consists in extracting them from the automatic speech recognition system (ASR) [START_REF] Canseco-Rodriguez | A comparative study using manual and automatic transcriptions for diarization[END_REF][START_REF] Tranter | Who really spoke when? Finding speaker turns and identities in broadcast news audio[END_REF][START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF][START_REF] Mauclair | Speaker diarization: about whom the speaker is talking?[END_REF]. The general principle is to determine if a detected named entity as a "PERSON" refers to a speaker of the document or to a person who does not speak in the document.

Our article takes place in this framework. The system we developed in [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF][START_REF] Mauclair | Speaker diarization: about whom the speaker is talking?[END_REF] uses uttered full names to assign them to anonymous speakers from identified speaker turns. The principle is to assign one of these four labels: "current turn", "previous turn", "following turn" or "another person" to each detected full name. But this approach ignored the potential conflict information on the speakers within a same speaker turn. In this paper, we propose to improve the consistency of the system and to better combine the various information on the potential speakers. The formalism of Belief Functions seemed to be particularly suited to managing these conflicts and combining this information.

First, we briefly present the automatic transcription system used, before describing the reference system for speaker named identification. Then, we discuss the shortcomings of this model and the improvements of our model using belief functions. Finally, we propose metrics for evaluating such systems, and we comment on the results obtained on the ESTER I evaluation campaign [START_REF] Galliano | The ESTER phase II evaluation campaign for the rich transcription of French broadcast news[END_REF].

2 Speaker identification based on a transcription system

Transcription system

The main hypothesis initially proposed in [START_REF] Canseco-Rodriguez | A comparative study using manual and automatic transcriptions for diarization[END_REF] assumes that a detected full name in a speaker turn allows to identify the current turn or a directly contiguous turn (previous or following turn). However, some full names identify farther speaker turns or persons that are not involved in the document. The used identification method is based on previously transcribed and enriched documents. This transcript needs to cut the document into segments which are then classified in anonymous speakers. These segments, grouped into speaker turns are transcribed and the named entities are annotated. The LIUM transcription system is described in [START_REF] Deléglise | The LIUM speech transcription system: a CMU Sphinx III-based system for French broadcast news[END_REF]. During the ESTER 1 (phase II) evaluation campaign in 2005 [START_REF] Galliano | The ESTER phase II evaluation campaign for the rich transcription of French broadcast news[END_REF], for the transcription task, our system was ranked second. This system achieves 20.5 % of word error rate on the evaluation corpus.

Semantic classification trees

The speaker identification method uses a binary decision tree based on the principle of semantic classification trees (SCT) [START_REF] Kuhn | The application of semantic classification trees to natural language understanding[END_REF]. A SCT automatically learns lexical rules from full names detected in the training corpus, with the left and right surrounding words. A SCT is used for each occurrence of full names detected in the transcripts. This tree allows to associate to each occurrence of a full name the probability to correspond to one of the four envisaged hypotheses:"current turn", "previous turn", "following turn" or "another person". These probabilities are determined during the learning of the tree and reflect the observed cases in the training corpus.

Reference combination method

The final goal of the system is to assign a full name to each anonymous speaker. Let us recall here the combination method of information provided by the trees that have been proposed in [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF]. Let E = {e 1 , . . . , e I } denotes the closed set of full names hypotheses to assign to a speaker. These candidates come from an exhaustive list of possible speakers known by the system. The set O = {o 1 , . . . , o J } corresponds to the successive occurrences of full names detected in the transcripts, T = {t 1 , . . . , t K } is the set of the speaker turns in chronological order, and C = {c 1 , . . . , c L } is the set of anonymous speakers to be labeled. Thus, the main goal is to assign a full name of E to a speaker of C. Each speaker c l may involve one or several times in a broadcast, that corresponds to several speaker turns: c l = {t ∈ T |c l is the speaker of turn t}. In a same turn, several occurrences of full names may be detected. For each occurrence of a full name o j (for j = 1, . . . , J) detected in a given speaker turn t k , let us define by P (o j , t k ) the probability that o j is current speaker. Thus, P (o j , t k-1 ) and P (o j , t k+1 ) represents the probability that o j is respectively the speaker of the previous and the following turn. By hypothesis, the probability that o j is another speaker is: 1 -1 r=-1 P (o j , t k+r ). At this stage, a filter is made by the comparison of genders: if the gender of the full name e i and the speaker c l are different, the corresponding occurrence is ignored. Let g(e i ) and g(c l ) be the respective gender (female, male or unknown) of e i and c l . The speaker gender is detected by the acoustic segmentation and classification system with high reliability and the full name gender is determined by the first name (generally without ambiguity) from a linguistic base of first names.

In [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF], to assign a full name e i to a speaker c l , we have computed a "score" for each full name e i , denoted as s l (e i ). This score is no more a probability, but is simply a sum of probabilities concerning the speaker turns of c l and taking gender constraints into account:

s l (e i ) = {(oj ,t)|oj =ei, t∈c l ,g(ei)=g(c l )} P (o j , t) (1)

Decision process

The goal is now to assign a full name e i to each speaker c l . Let f : C → E be the assignment function of full names to speakers. The principle of our solution, proposed in [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF], is actually to find a coherent matching between full names and speakers. Let D = {c l ∈ C| ∀e i ∈ E, s l (e i ) = 0}, be the set of speakers with no potential candidate. Several strategies may be used to sort the competing speakers c l for a given full name e i . The more natural way seems to choose the full name which has the maximum score for a given speaker e i (if there is at least one non-null score). Let us define the rule R 1 by:

∀c l ∈ C \ D, e * i = arg max ei∈E s l (e i ) ⇒ f (c l ) = e * i ∀c l ∈ D, f (c l ) = Anonymous. ( 2
)
An issue is that the same full name e * i may be assigned to several speakers. We proposed to reorganize the sharing of full names among speakers. Let the coefficient β il define the relative score of e i among all the possible candidates for assignment to c l :

β il = s l (ei) I q=1 s l (eq) if c l / ∈ D and β il = 0 if c l ∈ D. A concrete
example is given in Table 1. The full name "Jacques Derrida" has been assigned to three different speakers from the decision rule in Equation 2. In this example, c 13 has the best score, and "Jacques Derrida" should be assigned to c 13 ; but the score represents only 35% of the total scores among all the possible candidates for c 13 , whereas the score for c 15 represents 80% of total scores. Then, for the final decision, we have proposed to use the product of score s l (e i ), by coefficient

β il (rule R 2 ): SC l (e i ) = s l (e i )β il . (3)
Finally, in the same example, "Jacques Derrida" is assigned to c 15 and the speakers c 13 and c 14 will be labeled with other full names. The algorithm is iterative: all the full names are taken a priori into account and sorted according to their score SC l (e i ). First, the full name with the maximum score (denoted e * i ) is chosen, and if several speakers are associated to the same e * i , then e * i is assigned to the speaker whose score SC l (e * i ) is maximum. Then, all chosen full names are deleted from the list of speakers that are not yet assigned in this first iteration. In a second iteration, remaining full names are examined in the same way for the remaining speakers and so on, until all speakers are assigned, or their list is empty. Table 2 shows the result of this algorithm for the preceding example.

Drawbacks of the combination method

The combination method described above has several serious drawbacks, even though it has yielded good results [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF]. First, the concept of score is difficult to interpret, the quantities obtained in Equations 2 and 3 do not represent a degree of confidence, or a probability that a full name is a given speaker. They lead to a lack of clarity of the decision. Equation 3 represents a compromise that is difficult to justify. But the main drawback concerns the lack of uncertainty management in the combination method: particularly, conflict information in a given speaker turn is not taken into account. The available information is not correctly combined as a whole. No link is made between the different information provided by the classification tree, particularly when a same speaker pronounces several different full names and can therefore lead to erroneous results. Table 3 presents an example of a speaker turn t k where 8 occurrences are detected. The probabilities correspond to the next speaker turn t k+1 , who is a male. A female full name is therefore eliminated and two full names are rejected because they do not belong to the list. Some occurrences are redundant, because they correspond to a repeated full name and only one occurrence has a relative high probability. Two full names are still competing and they represent a significant incompatibility. These full names have high scores: Jean-Claude Pajak (1.25) and Jacques Chirac (0.87). These scores are close to those obtained if we had some information without ambiguity, for example a turn with only one occurrence with a high probability. This example highlights the fact that this method does not take into account the contradictory information provided by some speaker turns. A probabilistic formalism based on conditional probabilities could be considered for this kind of situation, but the lack of a priori information makes this type of modeling difficult. Even though the classification tree outputs are probabilistic, belief theory seemed more appropriate and less restrictive, particularly in the flexibility of its use.

Belief functions for speaker recognition

The contribution of this article lies in the combination process of different information, especially from the classification tree.

Belief function theory

In this section, we briefly recall some notions of the belief function theory [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF][START_REF] Smets | The transferable belief model[END_REF]. In this article, we adopt the point of view proposed by Smets: the Transferable Belief Model (TBM) [START_REF] Smets | The transferable belief model[END_REF]. The aim of this model is to determine the belief concerning different propositions, from some available information. Let Ω be a finite set, called frame of discernment of the experience. The representation of the uncertainty is made by the means of the concept of belief function, defined as a function m from 2 Ω to [0, 1] such as A⊆Ω m(A) = 1. The quantity m(A) represents the belief exactly allowed to proposition A. The subsets A of Ω such as m(A) > 0 are called the focal elements of m. One of most important operations in the TBM is the procedure for aggregating operator to combine several belief functions defined in a same frame of discernment [START_REF] Smets | The transferable belief model[END_REF]. In particular, the combination of two belief functions m 1 and m 2 "independently" defined on Ω using the conjunctive binary operator ∩, denoted as m = m 1 ∩ m 2 , is defined as [START_REF] Smets | The transferable belief model[END_REF]:

∀A ⊆ Ω, m (A) = B∩C=A m 1 (B)m 2 (C). (4)
Repeatedly, we may define the combination of n functions m 1 , . . . , m n on Ω by: m = m 1 ∩. . .∩m n . Once a belief function m is defined, it is possible to transform it into a probability distribution particularly for decision aspects. One of these, called pignistic probability and denoted by P m , is defined for all ω ∈ Ω as [START_REF] Smets | The transferable belief model[END_REF], if m(∅) = 1:

P m ({ω}) = A⊂Ω m(A) |A|(1 -m(∅)) δ A (ω), ( 5
)
where |A| denotes the cardinality of A, δ

A (ω) = 1 if ω ∈ A and δ A (ω) = 0 if ω / ∈ A .

Definition of belief masses

In this article, we propose to improve the system described in [START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF] by taking into account the coherence of the whole information provided by contiguous speaker turns. As we have seen before, in a same turn, several occurrences corresponding to different full names may be detected. First, we focus on a turn t k with n k occurrences and owing to speaker c l . Let n k+r be the number of occurrences for the previous turn (r = -1) and the following one (r = 1). Let {o k j,r }, with r = -1, 0, 1 and j = 1, . . . , n k+r , be the occurrences of the detected full names in these three turns. Each occurrence o k j,r , corresponding to a label e i , represents some knowledge concerning the speaker of the turn t k that can be described by a simple support belief function m jr t k on E, focused on e i and E:

m jr t k ({e i }) = α ij P (o k j,r , t k-r ) si o k j,r = e i m jr t k (E) = 1 -α ij P (o k j,r , t k-r ), (6)
where α il ∈ [0, 1] is a confidence measure of gender compatibility between e i and c l . If the genders are known with certainty, α il = 0 if g(e i ) = g(c l ) and α il = 1 if g(e i ) = g(c l ). If the first names are ambiguous (like Dominique in French) or unspecified, or if the speaker gender is uncertain, α il ∈]0, 1[ is estimated from a database of first names and the training corpus. Table 4 presents the belief function concerning the speaker of turn t k+1 in the example seen in section 2.5.

The belief mass of "Jean-Claude Pajak" is still high while the one of the other candidate is very low, and the degree of conflict is important since the mass of the empty set is high.

Combination by speaker

The first combination step consists in aggregating the whole information in a given speaker turn. The combination of the n k-1 + n k + n k+1 belief functions focused on the t k and obtained by Equation 6 is made with conjunctive non normalized Dempster rule (Equation 4), in order to ensure associativity and commutativity of the combination: we obtain a belief function m t k that represents a more synthetic knowledge of speaker identity provided in turn t k , defined by:

m t k = 1 r=-1 n k+r j=1 m jr t k . (7)
The second combination step consists in aggregating the results obtained by each speaker turn for the whole broadcast news. The more relevant and natural consists in keeping on combining all the belief functions focused on the same speaker c l with the same conjunctive Dempster rule and therefore combining all the belief functions corresponding to the speaker turns t k of this speaker. Thus, we obtain a global belief function M l which represents the state of belief concerning speaker c l for the whole broadcast news, and defined by:

M l = t k ∈c l m t k (8)

Decision rule

We use a similar procedure presented in section 2.4, but the decision process is simplified and unified thanks to the use of belief functions. We transform the belief function M l into the pignistic probability P M l (Equation 5) and we obtain the following rule R:

∀c l ∈ C \ D, e * i = arg max ei∈E P M l (e i ) ⇒ f (c l ) = e * i ∀c l ∈ D, f (c l ) = Anonymous. (9)
Then, since some full names may initially be assigned to several speakers, we apply the same decision process developed in 2.4, replacing scores SC l by pignistic probabilities P M l . If we come back to the proposed example in 2.4, the full name "Jacques Derrida" is again initially assigned to three speakers c 13 , c 14 and c 15 (see Table 5). Finally,"Jacques Derrida" is also assigned to c 15 , because this speaker has the most important pignistic probability. 4 Evaluation of the proposed system

Data description

The system evaluation are realized on French broadcast news records from the French ESTER 1 phase II evaluation campaign [START_REF] Galliano | The ESTER phase II evaluation campaign for the rich transcription of French broadcast news[END_REF]. The data were recorded from 5 French radios and Radio Télévision Marocaine and last from 10 to 60 minutes. They are divided in 3 corpora used for the SCT training, the system development and the evaluation: the training corpus of 76h (7416 speaker turns, 11292 detected full names), the development corpus of 30h (2931 speaker turns, 4533 full names) and the test corpus of 10h (1082 speaker turns, 1541 full names). This corpus contains two radios which are not present in the training and the development corpora. It was also recorded 15 months after the previous data.

Metrics

The results are evaluated comparing the generated hypothesis and the reference. This comparison highlights five cases:

-Identity is correct (C 1 ): the identity hypotheses corresponds to the correct one in the reference.

-Substitution error (S): the identity hypotheses differs from the one found in the reference. -Deletion error (D) : no identity is proposed although the speaker is identified in the reference. -Insertion error (I): an identity is proposed although the speaker is not identified in the reference. -No identity (C 2 ): no identity is proposed, and there is no identity for this speaker in the reference.

Among the measures defined in [START_REF] Tranter | Who really spoke when? Finding speaker turns and identities in broadcast news audio[END_REF][START_REF] Jousse | Automatic named identification of speakers using diarization and ASR systems[END_REF], the one that seems to best summarize the results is the global error rate Err computed from these 5 quantities:

Err = S + I + D S + I + D + C 2 + C 1 . (10)
The errors may be computed in terms of duration or in terms of number of speakers.

System evaluation

During experiments, the system is supposed to know all the full names that may be the speakers. This list is composed 1008 full names (the set E). Comparison between the reference system (c.f. [5] and section 2.3), and the proposed system is made on manual transcripts and segmentations. However, the named entities detection is automatic and may have some errors. The reference system is described in section 2.3 with two rules using scores s l (e i ) and SC l (e i ) (c.f. Equations 2 and 3) and our model is based on belief functions (Equation 9). As Table 6 shows, in the new model, the error rate in terms of duration (ErrDur) is 3 points less than reference system with rule R 2 and 7 points less with rule R 1 . Not only the use of belief functions is more easily interpretable, but also it allows to eliminate much errors. Concerning the number of identified speakers, the result is even more obvious:the new system correctly labels much more speakers than the base system, and also improves the reference system. In conclusion, taking account global information on speakers within a speaker turn, and before the decision, allows to significantly improve results both in terms of duration and in terms of numbers of speakers. Table 6. Comparison between the proposed system and the reference system according to the decision rule on the test corpus of ESTER 1 phase II campaign; ErrDur: error rate in duration; ErrSpk: error rate in number of speakers.

System

ErrDur ErrSpk Reference (rule R1) 20.6% 20.2% Reference (rule R2) 16.6% 19.5% Proposed (rule R) 13.7% 14.9%

Conclusion

The speaker identification method proposed in this article allows to extract speaker identities from transcriptions. The identification is realized thanks to a semantic classification tree which helps to give the full names found in the transcription to speakers in a recording. In this article, we propose a new system that consistently combines different information about the potential speakers in the form of belief functions. Particularly, the system manages possible conflict of information on the speakers within a speaker turn and takes into account the uncertainty concerning the gender. Experiments have been realized on a French broadcast news and the system have very good performances. Future work will focus on developing solutions to deal with automatic outputs containing errors. Different kind of uncertainty, dues to segmentation error, classification in speakers or to the bad transcription of full names will be taken into account. We will also study the realistic case of open systems when the list of possible speakers is unknown.

Table 1 .

 1 Example of an initial multiple assignment

	Speaker Full name e * i	s l (e * i ) β il SC l (e * i )
	c13 Jacques Derrida 8.58 35% 3.00
	c14 Jacques Derrida 1.67 56% 0.94
	c15 Jacques Derrida 4.94 80% 3.95

Table 2 .

 2 Example of the decision process with two iterations (decision in bold type, scores in parenthesis).

	Speaker e * i (1st iteration)	2nd iteration
	c13	J. Derrida (3.00) N. Demorand (0.25)
	c14	J. Derrida (0.94)	A. Adler (0.56)
	c15	J. Derrida (3.95)	-
	c16 O. Duhamel (1.15)	-

Table 3 .

 3 Score contribution in a speaker turn (t k+1 is a male).

	Occurrence oj	gender belongs to the list P (oj, t k+1 ) score
	Oscar Temaru	M	No	0.29	-
	Hamid Karzaï	M	No	0.29	-
	Jacques Chirac	M	Yes	0.29	0.87
	Jacques Chirac	M		0.29
	Jacques Chirac	M		0.29
	Jean-Claude Pajak	M	Yes	0.29
	Jean-Claude Pajak	M		0.96	1.25
	Véronique Rebeyrotte F	Yes	0.29	-

Table 4 .

 4 Mass distribution of the belief function in a speaker turn.

	Focal elements	mt k+1 ({ei})
	Jacques Chirac	0.018
	Jean-Claude Pajak	0.348
	∅	0.624
	E	0.010

Table 5 .

 5 Decision with two iterations (decision in bold, belief masses in parentheses).

	Speaker e * i (1st iteration)	2nd iteration
	c13	J. Derrida (0.89) N. Demorand (0.11)
	c14	J. Derrida (0.71)	A. Adler (0.25)
	c15	J. Derrida (0.99)	-
	c16 O. Duhamel (0.88)	-

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Anne-Gwénaëlle Guezennec
email: a.guezennec@brgm.fr

Sabrina Hedrich

Axel Schippers

Andreas Kamradt

Sabine Matys

Raymond Michels

Catherine Joulian

Françoise Bodenan

Patrick D'hugues

BIOLEACHING IN STIRRED TANK REACTORS TO PROCESS KUPFERSCHIEFER-TYPE ORE: CURRENT STATUS AND PERSPECTIVES

Keywords: bioleaching, copper, black shales, Kupferschiefer

Introduction

In Europe, most of the primary resources with high or moderate metal grade, reasonable accessibility and easy to process are exhausted. European primary resources still available for exploitation have more complex mineralizations (e.g. polymetallic and polymineral, carbon rich), or low metal contents; they also have higher levels of toxic impurities such as arsenic, antimony and mercury, penalizing current pyrometallurgical technologies. As existing processes and technologies are often not profitable for these types of unconventional resources, new process options still need to be developed in order to overwhelming the complexity of the ore composition while remaining cost effective. In this context, bioleaching is more and more considered as a promising technology. Even though heap and dump bacterial leaching of sulphidic minerals are well established and the bacterial treatment of refractory gold concentrate using stirred tank reactors (STR) is an industrial reality, the European mineral industry is still sceptical and reluctant to adopt biohydrometallurgical techniques. Heap leaching is often considered as un-adapted due to space constraints, slow leaching kinetics and low recovery rate. The possibility of using STR for the treatment of other metals than refractory gold, such as copper sulphides, has already been demonstrated but improvements are still needed to achieve economic viability.

The Kupferschiefer deposits host the largest known copper reserve in Europe [START_REF] Borg | An Overview of the European Kupferschiefer Deposits[END_REF]. These black shale type ores are currently exploited in Poland through pyrometallurgical smelting. In Germany exploration campaigns were recently leaded in order to assess and prepare future exploitation of this ore deposit type. The main copper-bearing minerals are: chalcocite, bornite, chalcopyrite and covellite. This type of ore is also characterized by high amounts of carbonate and organic carbon as well as potentially rich with arsenic (volatile in pyrometallurgical processes). In the last years the ores are characterized by increased As and C contents, and lower Cu contents. It leads to a lower quality concentrate as well as operating and environmental issues during smelting. In this context, several European research projects have been dedicated to the development of new bioleaching approaches as alternative and complementary routes to the conventional smelting methods for the processing of Kupferschiefer ores (BioShale in EC-FP6, ProMine in EC-FP7, BIOMOre). By using a multi-scale approach from molecular techniques to bench-scale small pilot continuous tests, Cu recovery from this type of ores using bioleaching was demonstrated to be technically feasible and efficient. The stirred tank bio-reactor (STR) was shown as the best process option when compared to heap leaching due to the high content of carbonate in the ore. This paper gives an overview of the work performed on this topic in the last decade. The R&D challenges necessary to tackle in order to improve the process economy will be analysed and compared. It will also discuss the new insights and future developments brought by ECOMETALS, a German and French joint research project, for the integration of bioprocess options in the metallurgical treatment of Kupferschiefer ores.

Selection and adaptation of microbial consortia

Several types of microbial consortia were tested: acidophilic autotrophic bacteria, acidophilic and neutrophilic heterotrophic bacteria. No evidence of efficient biodegradation was found with heterotrophic bacteria. Amongst the different types of autotrophic microorganisms used to bioleach Kupferschiefer ore, the best performances were obtained with the "BioShale-BRGM" consortium which is co-dominated by Acidithiobacillus (At.) caldus and Leptospirillum (L.) ferriphilum. The species Sulfobacillus (Sb.) benefaciens BRGM2 and Sb. thermosulfidooxidans are also present in lower ratios.

Bioleaching development in stirred tank reactor

Following successful cultures in batch tests, several continuous bioleaching operations were carried out using the consortium presented in the preceding section in a laboratory-scale unit equipped with three stirred reactors, one of 50 L (R1) followed by two of 20 L of operating capacity (R2, R3). The reactors are arranged in cascade so that the pulp flows from one tank to the next one by overflowing. CO 2 -enriched air (1%) is injected beneath the turbine at the bottom of the tank. The mixing system (BROGIM® -BRGM/MRM) is mounted in all tanks on a rotating shaft. In the reactors temperature was maintained constant at 42 °C and pH was regulated around 1.5. The agitation speed and the aeration flow were progressively slowed down (from 450 to 250 rpm and from 2000 to 100 L h -1 ) in order to decrease redox potential and improve chalcopyrite dissolution. The influence of the solid concentration was also studied (15% to 25% w/w).

Cu dissolution was positively impacted by the decrease of the agitation and aeration rate (Fig. 1). As can be seen in Fig. 2, Cu recovery is not affected by the increase of the solid load from 15% (w/w) to 25% (w/w). This solid content is quite high compared to those encountered in most of the commercial applications of bioleaching in CSTR (between 15 and 20%). No mixing or microbial issues were encountered. The Bioshale-BRGM consortium (which is mesophile to moderately thermophile) has shown a rare copper tolerance since copper content increased up to more than 40 g L -1 without any negative effect on the bacterial community.

Economic assessment and future developments

Based on the experimental data a preliminary techno-economic evaluation of the bioleaching option was established. The CAPEX is considerably decreased with a process design at higher solid load. This is mainly due to the reduction of the global bioreactor volume: operating at 25% solid load requires 2.3 less volume than operating at 15% solid load for the same Cu recovery and the same residence time. Following these calculations, elements on process economy were reviewed in order to point out the most pertinent ways to improve it. The highest investments costs generally come from bioleaching reactors and agitators. One option to decrease the CAPEX and to improve bioleaching process economy might be to develop new types of bioreactors as an alternative to classical CSTR. This new concept has already been proposed for other types of resources and is currently tested on Kupferschiefer ores in the German and French research project ECOMETALS [START_REF] Guezennec | Influence of dissolved oxygen on the bioleaching efficiency under oxygen enriched atmosphere[END_REF].

Figure 1 .Figure 2 .

 12 Figure 1. Influence of reduced agitation/aeration rate on copper dissolution


Mogens Fosgerau

Emerson Melo
email: emelo@iu.edu

André De Palma
email: andre.depalma@ens-cachan.fr.

Matthew Shum
email: mshum@caltech.edu

Bob Becker

Marcus Berliant

Mark Dean

Federico Echenique

Juan Carlos Escanciano

Filip Matejka

Paulo Natenzon

Antonio Rangel

Ryan Webb

Michael Lucie Woodford

Julien Letrouit

Alejandro Monardo

Robin- Son Cortes

Discrete Choice and Rational Inattention: a General Equivalence Result *

Keywords: JEL codes: D03, C25, D81, E03 Keywords: Rational Inattention, discrete choice, random utility, convex analysis, generalized entropy

This paper establishes a general equivalence between discrete choice and rational inattention models. Matejka and McKay (2015, AER) showed that when information costs are modelled using the Shannon entropy function, the resulting choice probabilities in the rational inattention model take the multinomial logit form. By exploiting convex-analytic properties of the discrete choice model, we show that when information costs are modelled using a class of generalized entropy functions, the choice probabilities in any rational inattention model are observationally equivalent to some additive random utility discrete choice model and vice versa. Thus any additive random utility model can be given an interpretation in terms of boundedly rational behavior. This includes empirically relevant specifications such as the probit and nested logit models.

Motivation

In many situations where agents must make decisions under uncertainty, information acquisition is costly (involving pecuniary, time, or psychic costs); therefore, agents may rationally choose to remain imperfectly informed about the available options. This idea underlies the theory of rational inattention, which has become an important paradigm for modeling boundedly rational behavior in many areas of 1 economics [START_REF] Sims | Implications of Rational inattention[END_REF](Sims ( , 2010))). In this paper, our main contribution is to establish a general equivalence between additive random utility discrete choice and rational inattention models. Matejka and McKay (2015) showed that when information costs are modelled using the Shannon entropy function, the resulting choice probabilities in the rational inattention model take the multinomial logit (MNL) form. In order for the rational inattention model to generate non-MNL choice probabilities, we need to generalize the information cost function beyond the Shannon entropy function assumed in much of the existing literature. We do this by exploiting convex-analytic properties of the additive random utility model to demonstrate a duality between discrete choice and rational inattention models. 1Specifically, we introduce a class of Generalized Entropy Rational Inattention (GERI) models. 2 In GERI models, the information cost functions are constructed from a class of "generalized entropy" functions; these generalized entropy functions are, essentially, "dual" to the class of random utility discrete choice models; precisely, the generalized entropy functions are the convex conjugate functions to the surplus functions in any arbitrary general additive random utility model. Hence, GERI models naturally yield choice probabilities that can equivalently be generated from general additive random utility models; the resulting choice probabilities can take forms far beyond the multinomial logit, including specifications such as nested logit, multinomial probit, and so on, which are often employed in empirical work.

Importantly, these generalized entropy functions allow for random utility models in which the random shocks are dependent across options; this corresponds to information cost functions that exhibit information spillovers across options with shared features, which may be reasonable in many decision environments. In contrast, the multinomial logit model assumes independent shocks; correspondingly, the Shannon entropy function precludes information spillovers.

The paper is organized as follows. Section 2 presents insights into the fundamental convex-analytic structure of the additive random utility discrete choice model. Using this structure, we formulate a class of generalized entropy functions and present key results about them. Section 3 introduces the rational inattention model. We show how the generalized entropy functions can be used to define the information cost function in the rational inattention model. Then we present the key result from this paper, which establishes the equivalence between choice probabilities emerging from the discrete choice model, and those emerging from the rational inattention model based on the generalized entropy functions. Section 4 discusses an example while Section 5 concludes.

Notation: Throughout this paper, for vectors a and b, we use the notation a • b to denote the vector scalar product i a i b i . ∆ denotes the unit simplex in R N .

Random utility models and generalized entropy functions

Consider a decision-maker (DM) making discrete choices among a set of i = 1, . . . , N options, where, for each option i, the utility is given by

u i = ṽi + i , (1)
where ṽ = (ṽ 1 , . . . , ṽN ) is deterministic and = ( 1 , . . . , N ) is a vector of random utility shocks. This is the classic additive random utility framework pioneered by [START_REF] Mcfadden | Modelling the choice of residential location[END_REF].

Assumption 1 The random vector = ( 1 , . . . , N ) follows a joint distribution with finite means that is absolutely continuous, independent of ṽ, and fully supported on R N .

An important concept in this paper is the surplus function of the discrete choice model (so named by [START_REF] Mcfadden | Econometric Models of Probabilistic Choice[END_REF], defined as

W (ṽ) = E (max i [ṽ i + i ]).
(2)

Under Assumption 1, W (ṽ) is convex and differentiable3 and the choice probabilities coincide with the derivatives of W (ṽ): ∂W (ṽ) ∂ṽ i = q i (ṽ) ≡ P (ṽ i + i ≥ ṽj + j , ∀j = i) for i = 1, . . . , N or, using vector notation, q(ṽ) = ∇W (ṽ). This is the Williams-Daly-Zachary theorem in the discrete choice literature [START_REF] Mcfadden | Modelling the choice of residential location[END_REF][START_REF] Mcfadden | Econometric Models of Probabilistic Choice[END_REF]). As a running example, we consider the familiar logit model. When the i 's are distributed i.i.d. across options i according to the type 1 extreme value distribution, then the resulting choice probabilities take the familiar multinomial logit form: q i (ṽ) = e ṽi / j e ṽj . Assumption 1 above leaves the distribution of the 's unspecified, thus allowing for choice probabilities beyond the multinomial logit case. Importantly, it accommodates arbitrary correlation in the i 's across choices, which is reasonable and realistic in applications.

We define a vector-valued function

H(•) = (H 1 (•), ..., H N (•)) : R N + → R N
+ as the gradient of the exponentiated surplus, i.e.

H(e ṽ) = ∇ ṽ e W (ṽ) .

(3)

From the differentiability of W and the Williams-Daly-Zachary theorem it follows that the choice probabilities emerging from any random utility discrete choice model can be expressed in closed-form in terms of the H function as:4 q i (ṽ) =

H i e ṽ N j=1 H j (e ṽ)

, for i = 1, . . . , N .

For the multinomial logit case, the surplus function is W (ṽ) = log N i=1 e ṽi , implying that H i (e ṽ) = e ṽi . Thus, for this case Eq. ( 4) becomes the multinomial logit choice formula.

The function H is globally invertible (see Lemma 8 in the Appendix), and we introduce a function S defined as the inverse of H,

S(•) = H -1 (•).
(5)

For reasons that will be apparent below, we refer to S as a generator function.

There is a close relationship between the function S(•) and the surplus function W (ṽ) of the corresponding discrete choice model: as the next proposition establishes, the surplus function W (•) and the generator function S(•) are related in terms of convex conjugate duality (Rockafellar, 1970, ch. 12).5

Proposition 1 (Convexity properties and generalized entropy functions) Let assumption 1 hold. Then:

(i) The surplus function W (ṽ) is equal to

W (ṽ) = log N i=1 H i (e ṽ) . (6)
(ii) The convex conjugate function for the surplus function W (ṽ) is

W * (q) = q • log S(q) q ∈ ∆ +∞ otherwise,
where S(•) is a generator function defined in (5). We call the negative convex conjugate function -W * (•) a generalized entropy function.

(iii) The surplus function W (ṽ) is the convex conjugate of W * (q), that is

W (ṽ) = max q∈∆ {q • ṽ -W * (q)} (7)
and the RHS is optimized at the choice probabilities q(ṽ) = ∇W (ṽ).

Parts (i) and (ii) establish a specific structure of the surplus function W and its convex conjugate W * ; this is new in the literature on random utility models, and may be of independent interest. We use this structure to define the class of generalized entropy functions. To see how this works, consider again the multinomial logit model, for which H is the identity, implying that the corresponding generator function S(q) = q is also just the identity. Then by Proposition 1(ii), the negative convex conjugate function is -W * (q) = -q • log q =i q i log q i , which is just the [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] entropy function.

Generalizing from this, Proposition 1(ii) shows how the conjugate function for any discrete choice model can be generated by the function S. Therefore we refer to the negative conjugate function -W * (q) = -q • log S(q) =i q i log S i (q) for any general discrete choice model as a generalized entropy function. Comparing the generalized and Shannon entropies, the former allows for cross-effects, in the sense that the choice probability for option j, q j , enters the entropic term for option i, S i (q). As we will see below, these cross-effects allow for information spillovers when we use these generalized entropy functions to construct rational inattention models.

Proposition 1(iii) provides an alternative representation of the surplus function from a random utility model, in addition to Eq. ( 2). It illustrates a close connection between -W * (q) and the joint distribution of , the random utility shocks, which aids interpretation of the generalized entropy function. Specifically, Eq. (2) implies that the surplus function can be written as

W (ṽ) = N i=1 q i (ṽ)(ṽ i + E( i |u i ≥ u j , j = i)).
Combining this with (7), we obtain an alternative expression for the generalized entropy function, as a choice probability-weighted sum of expectations of the utility shocks :6

-W * (q) = i q i E[ i |u i ≥ u j , j = i].
In this way, different distributions for the utility shocks in the random utility model will imply different generalized entropy functions, and vice versa.

We conclude this section enumerating some properties of the generator functions S(•), which will be important in what follows.

Proposition 2 (Properties of the generator functions) Let assumption 1 hold. Then the vector valued-function S(•) defined by (5) satisfies the following conditions:

(i) S is continuous and homogenous of degree 1.

(ii) q • log S(q) is convex.

(iii) S is differentiable with :

N i=1 q i ∂ log S i (q) ∂q k = 1, k ∈ {1, . . . , N },
where q is a probability vector with 0 < q i < 1 for all i.

The possibility of zero choice probabilities will play a role in what follows. We impose an additional regularity assumption on the generator functions S.

Assumption 2 Let q be a probability vector. Then q i = 0 if and only if S i (q) = 0.

This assumption is satisfied for the generator functions S corresponding to many familiar additive random utility models, including the multinomial logit and the nested logit models.7

Rational inattention

We now introduce the rational inattention model. The decision maker is again presented with a group of N options, from which he must choose one. Each option has an associated payoff v = (v 1 , ..., v N ), but in contrast to the additive random utility model, the vector of payoffs is unobserved by the DM. Instead, the DM considers the payoff vector V to be random, taking values in a set V ⊂ R N ; for simplicity, we take V to be finite. The DM possesses some prior knowledge about the available options, given by a probability measure µ(v) = P(V = v).

The DM's choice is represented as a random action A that is a canonical unit vector in R N . The payoff resulting from the action is V•A, namely that value of the entry in V indicated by the action A. The problem of the rationally inattentive DM is to choose the conditional distribution P(A|V), balancing the expected payoff against the cost of information.

Denote an action by i and write p i (v) as shorthand for P (A = i|V = v). Denote also the vector of choice probabilities conditional on V = v by p(v) = (p 1 (v), . . . , p N (v)), and let p(•) = {p(v)} v∈V denote the collection of conditional probabilities. The DM's strategy is a solution to the following variational problem: max

p(•) (E (V • A) -information cost) . (8)
The previous literature has used the Shannon entropy to specify the information cost, which connects the rational inattention model to the multinomial logit model. We review these results in the next Section 3.1. Then in Section 3.2 we introduce generalized entropy to the problem. This connects the rational inattention model to general additive random utility models.

Shannon entropy and multinomial logit

The key element in the program above is the cost of information. Much of the previous literature has utilized the mutual (Shannon) information between payoffs V and the actions A to measure the information costs. Denote the Shannon entropy by Ω(q) = -q • log q. Denote also the unconditional choice probabilities by p 0 i = Ep i (V) and p 0 = (p 0 1 , . . . , p 0 N ). Then the mutual (Shannon) information between V and A may be written as

κ(p (•) , µ) = Ω(E(p(V))) -E(Ω(p(V))) (9) = - N i=1 p 0 i log p 0 i + v∈V N i=1 p i (v) log p i (v) µ(v). (10)
Accordingly, we can specify the information cost as λκ(p, µ) where λ > 0 is the unit cost of information. As the distribution of payoffs is unspecified, we may take λ = 1 at no loss of generality. The choice strategy of the rationally inattentive DM is the distribution of the action A conditional on the payoff V that maximizes the expected payoff less the cost of information, which is the solution to the optimization problem

max p(•) v∈V N i=1 v i p i (v) µ(v) -κ(p (•) , µ) (11)
subject to

p i (v) ≥ 0 for all i, N i=1 p i (v) = 1. (12)
Solving this, the DM finds conditional choice probabilities

p i (v) = p 0 i e v i N j=1 p 0 j e v j for i = 1, . . . , N , (13)
that satisfy p 0 i = Ep i (V). It is an important feature of the rational inattention model that some p 0 i may be zero, in which case the corresponding p i (v) are also zero. Then the rational inattention model implies the formation of a consideration set, comprising those options that have strictly positive probability of being chosen (cf. [START_REF] Caplin | Rational Inattention, Optimal consideration sets and stochastic choice[END_REF]).

Under the convention that log 0 = -∞ and exp (-∞) = 0, we may rewrite (13) as

p i (v) = e v i +log p 0 i N j=1 e v j +log p 0 j = e ṽi N j=1 e ṽj ,
where ṽi = v i + log p 0 i . This may be recognized as a multinomial logit model in which the payoff vector ṽ is v shifted by log p 0 . For options that are not in the consideration set, the shifted payoff is ṽi = -∞. From the perspective of the multinomial logit model these options have zero probability of maximizing the random utility (1) and they have effectively been eliminated from the model.

The Generalized Entropy Rational Inattention (GERI) model

In this paper we generalize the preceding equivalence result between rational inattention and multinomial logit. To achieve that, we replace the Shannon entropy by the generalized entropy introduced in Section 2 above. Since each generalized entropy implies a corresponding discrete choice model (Proposition 2), it turns out that each RI model with an information cost derived from a generalized entropy will generate choice probabilities consistent with a corresponding discrete choice model (Proposition 4 below); this implies that any additive random utility discrete choice model can be microfounded by a corresponding rational inattention model, thus generalizing substantially the results in the previous section.

We begin by generalizing the rational inattention framework described above, using generalized entropy in place of the Shannon entropy. Specifically, we let S be the entropy generator corresponding to some additive random utility model satisfying Assumptions 1 and 2, and define Ω S (p) = -p • log S (p) as the corresponding generalized entropy. We define accordingly a general information cost by

κ S (p (•) , µ) = Ω S (Ep(V)) -EΩ S (p(V)) (14) = -p 0 • log S p 0 + v∈V [p (v) • log S (p (v))] µ (v) .
A Generalized Entropy Rational Inattention (GERI) model describes a DM who chooses the collection of conditional probabilities p (•) = {p(v)} v∈V to maximize his expected payoff less the general information cost

max p(•) v∈V N i=1 v i p i (v) µ(v) -κ S (p (•) , µ). ( 15
)
The following proposition characterizes the optimal solution to the GERI model.

Proposition 3

The solution to the GERI model:

(i) The unconditional probabilities satisfy the fixed point equation

p 0 = E   H e V+log S(p 0 ) N j=1 H j e V+log S(p 0 )   . ( 16
)
(ii) The conditional probabilities are given in terms of the unconditional probabilities by

p i (v) = H i e v+log S(p 0 ) N j=1 H j e v+log S(p 0 ) . ( 17
)
(iii) The optimized value of ( 15) is

E log N j=1
H j e V+log S(p 0 ) = EW V + log S p 0 .

Part (i) of the proposition shows that the solution of the GERI model involves a fixed point problem; in what follows, we assume that a solution exists. Part (iii) illustrates the close connection between convex analysis and the GERI problem. To see this, note that the GERI information cost function may be written as

κ S (p(•), µ) = -W * (p 0 ) + EW * (p(V)). (18)
Hence, given p 0 , the conditional choice probabilities p(v) can be generated, for each v ∈ V, by the problem

max p(v)∈∆ p(v) • (v + log S(p 0 )) -W * (p(v)) , (19)
the optimized value of which, by Proposition 1(iii), is

W (v + log S(p 0 )), for each v ∈ V (20) corresponding to Proposition 3(iii).
It is worth remarking that some of the optimal unconditional choice probabilities may be zero. For these options, the corresponding conditional choice probabilities will also be zero for all v. 8 The rational inattention model then also describes the formation of consideration sets, i.e. the set of options that are chosen with positive probability. 9 While Proposition 3 does not characterize explicitly the optimal consideration set emerging from a GERI model, the following corollary describes one important feature that it has, namely that it excludes options that offer the lowest utility in all states of the world.

Corollary 4 For some option a, and for all v ∈ V, let v a ≤ v i for all i = a, and assume that the inequality is strict with positive probability. Then p 0 a = 0 (that is, option a is not in the optimal consideration set).

For the special case of Shannon entropy (when S is the identity function), the result can be strengthened even further. Corollary 7 in the Appendix shows that in that case, an option that is dominated by another option in all states of the world will not be in the optimal consideration set.

Equivalence between discrete choice and rational inattention

We now establish the central result of this paper, namely the equivalence between additive random utility discrete choice models and rational inattention models. In particular, we show that the choice probabilities generated by a GERI model lead to the same choice probabilities as a corresponding additive random utility model and vice versa.

Combining the choice probabilities p i (v) in ( 17) from the GERI model and the choice probabilities q i (ṽ) in (4) from the additive random utility model, we find that if payoffs are related by

ṽi = v i + log S i (p 0 ) for i = 1, . . . , N , (21)
then the two models yield the same choice probabilities

p i (v) = H i (e v+log S(p 0 ) ) N j=1 H j (e v+log S(p 0 ) ) = H i (e ṽ) N j=1 H j (e ṽ)
= q i (ṽ).

Given a GERI model with payoffs v ∈ V and unconditional choice probabilities p 0 , we may then use (21) to construct deterministic utility components ṽ for the additive random utility model. If the GERI model has some zero unconditional choice probabilities p 0 i , then Assumption 2 ensures that p i (v) = 0 if and only if q i (ṽ) = 0. The additive random utility model that corresponds to the GERI model is then an extended additive random utility model in which some deterministic utility components are minus infinity.

Conversely, given an additive random utility model with flexible generator S and a prior distribution μ of the deterministic utility components ṽ ∈ Ṽ, define p 0 = Eq(ṽ) and note that all p 0 i > 0. Then define v using (21) and define similarly µ and V using the same location shift log S(p 0 ). By the same argument as before, the GERI model with payoffs v ∈ V, prior µ and flexible generator S for the generalized entropy has the same choice probabilities as the additive random utility model.

Hence, we have shown the following proposition.

Proposition 5 For every additive random utility discrete choice model and every prior distribution on Ṽ, there is an equivalent GERI model with a prior distribution on V, where V is equal to Ṽ up to a location shift.

Conversely, every GERI model is equivalent to an additive random utility discrete choice model in which the utility components for options chosen with zero probability are set to minus infinity.

In Section 4, we will apply this proposition to study a GERI model in which the choice probabilities are equivalent to those from a nested logit discrete choice model.

Additional properties of generalized entropy cost functions

We have shown that the generalized rational inattention model is always equivalent to an additive random utility model and conversely that the generalized rational inattention model may provide a boundedly rational foundation for any additive random utility model. The key to this result is the generalization of the information cost function κ S (p (•) , µ) using generalized entropy as defined in Eq. ( 14). It is then natural to ask whether κ S (p (•) , µ) has the properties that one would desire for an information cost. In this section we show that κ S (p (•) , µ) does indeed possess two reasonable and desirable properties of cost functions that have been discussed in the existing literature (cf. [START_REF] Oliveira | Rationally Inattentive Preferences and Hidden Information Costs[END_REF], [START_REF] Hébert | Rational Inattention with Sequential Information Sampling[END_REF]), thus providing normative support for the GERI framework.

First, when A and V are independent, then the action A carries no information about the payoff V. In that case the information cost should be zero, i.e.

Independence. If A and V are independent, then κ S (p(•), µ) = 0.

Second, the mutual Shannon information κ(p (•) , µ) is a convex function of p. This is useful as it ensures a unique solution to the problem of the rationally inattentive DM. We show that the information cost κ S (p (•) , µ) has a slightly weaker property, namely that it is convex on sets where Ep(V) is constant.

Convexity. For a given µ, the information cost function κ S (p (•) , µ) is convex on any set of choice probabilities vectors satisfying {p : V → ∆| Ep(V) = p}.

The mutual Shannon information κ(p (•) , µ) satisfies these two properties. The next proposition establishes that the information cost defined in (14) using the generalized entropy functions also satisfies these properties.

Proposition 6 The information cost defined in Eq. ( 14) satisfies the independence and convexity conditions.

Example: The nested logit GERI model

From an applied point of view, an important implication of Proposition 5 is that it allows us to formulate rational inattention models that have complex substitution patterns, beyond the multinomial logit case. In this example, we consider a GERI model with an information cost function derived from a nested logit discrete choice model. The nested logit choice probabilities are consistent with a discrete choice model in which the utility shocks are jointly distributed in the class of generalized extreme value distributions. Among applied researchers, the nested logit model is often preferred over the multinomial logit model because it allows some products to be closer substitutes than others, thus avoiding the "red bus/blue bus" criticism. 10We partition the set of options i ∈ {1, . . . , N } into mutually exclusive nests, and let g i denote the nest containing option i. Let ζ g i ∈ (0, 1] be nest-specific parameters. For a valuation vector ṽ, the nested logit choice probabilities are given by: The S function corresponding to a nested logit model is

q i (ṽ) = e ṽi /
S i (q) = q ζg i i   j∈g i q j   1-ζg i (23)
Using this, and applying Proposition 5, the nested logit choice probabilities ( 22) are also equivalent to those from a GERI model with valuations

v i = ṽi -ζ g i log p 0 i -(1 -ζ g i ) log   j∈g i p 0 j   , i ∈ {1, . . . , n} . (24)
The S function for the nested logit model in Eq. ( 23) has several interesting features, relative to the Shannon entropy. First, Eq. ( 23) allows us to write the generalized entropy Ω S (p) as

Ω S (p) = - N i=1 ζ g i p i log p i - N i=1 (1 -ζ g i )p i log   j∈g i p j   . ( 25
)
The first term in Eq (25) captures the Shannon entropy within nests, whereas the second term captures the information between nests. According to this, we may interpret Eq. ( 25) as an "augmented" version of Shannon entropy. Second, when the nesting parameter ζ g j = 1, then S is the identity function (S j (p) = p j for all j), corresponding to the Shannon entropy. When ζ g j < 1, then S j (p) ≥ p j ; here, S(p) behaves as a probability weighting function which tends to overweight options j belonging to larger nests. At the extreme ζ g j → 0, all options within the same nest effectively collapse into one aggregate option and become perfect substitutes.

From the discrete choice perspective, nested logit choice probabilities allow for correlation in the utility shocks ( 's) corresponding to the different choice options. Analogously, in an information cost function constructed from the nested logit S function in Eq. ( 23), there will be a common cost component across all options belonging to the same nest, corresponding to the term ( j∈g j p j ) 1-ζg j which is common to all S j (p) for j ∈ g j . From an information processing perspective, this suggests that there are spillovers in gathering information for options in the same nest. Information spillovers across choices arise in many decision environments. For example, a supermarket shopper gains information about common features of the vegetables, such as the average freshness and quality, while looking at any of them. In animal foraging, animals who have information about presence of predators in one grazing site also use that information to update about predator presence at other nearby sites.

For the Shannon entropy, in contrast, these common terms do not exist, so that there are no spillovers across options in information processing. From a behavioral point of view, then, more correlated utility shocks makes each option's signal harder to distinguish -there is more redundant information -implying that multinomial logit choice probabilities, which would ignore this correlation, manifest a type of correlation neglect.

To illustrate this point, we compute a GERI model using the nested-logit cost function. (This requires solving the fixed point equation ( 16).) In this example, there are five options, in which the valuations v = (v 1 , v 2 , . . . , v 5 ) are drawn i.i.d. uniformly from the unit interval. We assume that options (1,2,3) are in one nest, and options (4,5) are in a second nest. With this specification, all five options are a priori identical, and have equal probability of being the option with the highest valuation. Hence, we might expect that any non-uniform choice probabilities should reflect underlying asymmetries in the information cost function.

In Table 1, we report the average choice probability for each option according for several specifications of the nested logit cost function. In the top panel, we set ζ 1 = ζ 2 = 1, corresponding to the multinomial logit model. In the bottom panel, we set

ζ 1 = ζ 2 = 0.5.
As we expect, we see that the average choice probabilities are identically equal to 0.2 across all five options in the multinomial logit case. As we remarked before, this reflects the feature of the Shannon-based information cost function (S i (p) = p i ) in which information costs are separable across all five choices. 11 Unlike the multinomial logit case, we see that choice probabilities are higher for the options Choice probs: Option 1 Option 2 Option 3 Option 4 Option 5

Multinomial logit: Moreover, the performance of the two models is surprisingly different. Under the multinomial logit specification, the overall efficiency -defined as the average probability of choosing the option with the highest valuation -is 28%. The overall efficiency for the nested logit, however, is higher, being over 35%.

ζ 1 = 1, ζ 2 =
This simple example demonstrates the substantive importance of allowing for information cost functions beyond the Shannon entropy, which leads to multinomial logit choice probabilities. Obviously, it makes a difference for a DM to be processing information using the nested logit cost function, as compared to the Shannon cost function, as the highest valuation option is chosen with higher probability on average using the nested logit cost function.

Summary

The central result in this paper is the observational equivalence between a random utility discrete choice model and a corresponding Generalized Entropy Rational Inattention (GERI) model. Thus the choice probabilities of any additive random utility discrete choice model can be viewed as emerging from rationally inattentive behavior, and vice-versa; we can go back and forth between the two paradigms. 12 Then, in order to apply an additive random utility discrete choice model, it is no longer necessary to assume that decision makers are completely aware of the valuations of all the available options. This is important, as it is clearly unrealistic to expect that decision makers to be aware of all options in a large set of options.

The underlying idea is that, by exploiting convex analytic properties of the discrete choice model, we show a "duality" between the discrete choice and GERI models in the sense of convex conjugacy. Precisely, the surplus function of a discrete choice model has a convex conjugate that is a generalized entropy. Succinctly, then, GERI models are rational inattention problems in which the information cost functions are built from the convex conjugate functions of some additive random utility discrete choice model.

A few remarks are in order. First, the equivalence result in this paper is at the individual level, hence it also holds for additive random utility models with random parameters, including the mixed logit or random coefficient logit models which have been popular in applied work. 13 Any mixed discrete choice model such as these is observationally equivalent to a mixed GERI model.

In addition, there is also a connection between the results here and papers in the decision theory literature. The GERI optimization problem (15) bears resemblance to the variational preferences that [START_REF] Maccheroni | Ambiguity Aversion, Robustness, and the Variational Representation of Preferences[END_REF] propose to represent ambiguity averse preferences, as well as to the revealed perturbed utility paradigm proposed by [START_REF] Fudenberg | Stochastic Choice and Revealed Perturbed Utility[END_REF] to model stochastic choice behavior. [START_REF] Gul | Random Choice as Behavioral Optimization[END_REF] shows an equivalence between random utility and an "attribute rule" model of stochastic choice. The main point in this paper is to establish a duality between rational inattention models and random utility discrete choice models, which results in observational equivalence of their choice probabilities. A similar duality might arise between random utility discrete choice models and these other models from decision theory.

Finally, there are rational inattention models outside the GERI framework; that is, rational inattention models with information cost functions outside the class of generalized entropy functions introduced in this paper. 14 Obviously, choice probabilities from these non-GERI models would not be equivalent to those which can be generated from random utility discrete-choice models; it will be interesting to examine the empirical distinctions that non-GERI choice probabilities would have.

C. Sims (2010). Rational inattention and monetary economics. Handbook of Monetary Economics, Volume 3, pp. 155-181. L. Thurstone (1927). A law of comparative judgment. Psychological review, 34(4): 273-278. R. Webb (2016). The Dynamics of Stochastic Choice. Working paper.

A Proofs and additional results

Notation. Vectors are denoted simply as q = (q 1 , ..., q N ). A univariate function applied to a vector is understood as coordinate-wise application of the function, e.g., e q = (e q 1 , ..., e q N ). Consequently, if a is a real number then a + q = (a + q 1 , ..., a + q J ). The gradient with respect to a vector ṽ is ∇ ṽ; e.g., for ṽ = (v 1 , ..., v N ), ∇ ṽW (ṽ) = ∂W (ṽ) ∂ṽ 1 , ..., ∂W (ṽ) ∂ṽ N

. The Jacobian is denoted J with, for example,

J log S (q) =    ∂ log S 1 (q) ∂q 1 ... ∂ log S 1 (q) ∂q N ... ... ... ∂ log S N (q) ∂q 1 ... ∂ log S N (q) ∂q N    .
A dot indicates an inner product or products of vectors and matrixes. For a vector q, we use the shorthand 1 • q = i q i . The unit simplex in R N is ∆.

Proof of proposition 1. We first evaluate W * (q). If 1 • q = 1, then q • (ṽ + γ) -W (ṽ + γ) = q • ṽ -W (ṽ) + (1 • q -1) γ, which can be made arbitrarily large by changing γ and hence W * (q) = ∞. Next consider q with some q j < 0. W (ṽ) decreases towards a lower bound as v j → -∞. Then q • ṽ -W (ṽ) increases towards +∞ and hence W * is +∞ outside the unit simplex ∆.

For q ∈ ∆, we solve the maximization problem

W * (q) = sup ṽ {q • ṽ -W (ṽ)}. (26)
Note that for any constant k we have W (ṽ + k • 1) = k + W (ṽ), so that we normalize 1 • ṽ = 0. Maximize then the Lagrangian q • ṽ -W (ṽ) -λ (1 • ṽ) with first-order conditions 0 = q j -∂W (ṽ) ∂ṽ j -λ, which lead to λ = 0. Then q = ∇ ṽW (ṽ) ⇔ qe W (ṽ) = ∇ ṽ e W (ṽ) = H e ṽ ⇔ S (q) e W (ṽ) = e ṽ ⇔ log S (q) + W (ṽ) = ṽ ⇒ q • log S (q) + W (ṽ) = q • ṽ.

Inserting this into (26) leads to the desired result.

W is convex and closed and hence W is the convex conjugate of W * (Rockafellar, 1970, Thm. 12.2). This, along with Fenchel's equality (Rockafellar, 1970, Thm. 23.5), proves part (iii). Finally, for part (i), let q be a solution to problem (7). Then, by the homogeneity of H we have q = 1 α H(e ṽ), where α = N j=1 H j (e ṽ). Then, by the definition of S it follows that S(q) = e ṽ α . Replacing the latter expression in Eq. ( 7) we get W (ṽ) = qṽq log e ṽ/α , = qṽq log e ṽ + log α ,

= log   N j=1 H j (e ṽ)   .
Proof of Proposition 2. Continuity of S follows from continuity of the partial derivatives of W , which is immediate from the definition. Homogeneity of S is equivalent to homogeneity of H. Using the homogeneity property of W S -1 (λe ṽ) = ∇ v (e W (ṽ+log λ) ) = λ∇ v (e W (ṽ) ) = λS -1 (e ṽ), which shows that H and hence S are homogenous of degree 1.

The requirement that N i=1 q i ∂ log S i (q) ∂q k = 1 in the relative interior of the unit simplex ∆ may be expressed in matrix notation as (q 1 , . . . , q N ) • J log S (q) = (1, . . . , 1), where

J log S (q) = ∂ log S i (q) ∂q j N i,j=1
is the Jacobian of log S(q). Defining t ≡ log S(q), we have q = H e t and hence W e t = log(1 •

H(e t)) = log 1 = 0 by Proposition 1. Noting that (log(S)) -1 ( t) = H(e t) the requirement in part (ii) is equivalent to (q 1 , . . . , q N ) = (q 1 , . . . , q N ) • J log S (q) • J (log S) -1 ( t) = (1, . . . , 1) • J H(e t) ( t).

Now, use the Williams-Daly-Zachary theorem to find that

(1, . . . , 1) • J H(e t) ( t) = ∇ t e W ( t) = e W (ṽ) (q 1 , . . . q N ) = (q 1 , . . . q N ) .

as required.

Part (ii) follows from Proposition 1(ii).

Proof of proposition 3. The Lagrangian for the DM's problem is

Λ = E (V • A)-κ S (p, µ)+E   γ (V)   1 - j p j (V)     +E   j ξ j (V) p j (V)   ,
where γ (V) and ξ j (V) are Lagrange multipliers corresponding to condition (12). Before we derive the first-order conditions for p j (v) it is useful to note that we may regard the terms log S p 0 and log S (p (v)) in the information cost κ S (p, µ) as constant, since their derivatives cancel out by Proposition 2(iii). Define ṽj = v j + ξ j (v) + log S j p 0 and ṽ = (ṽ 1 , ..., ṽN ). Then the first-order condition for p j (v) is easily found to be log S j (p (v)) = ṽj -γ (v) .

(

) 27
This fixes p (v) as a function of p 0 since then

p (v) = H e ṽ exp (-γ (v)) . (28)
If some p j (v) = 0, then we must have ṽj = -∞, which implies that S j p 0 = 0 and the value of ξ j (v) is irrelevant. If p j (v) > 0, then ξ j (v) = 0. We may then simplify by setting ξ j (v) = 0 for all j, v at no loss of generality, which means that ṽj = v j + log S j p 0 .

Using that probabilities sum to 1 leads to exp (γ (v)) = j H j e ṽ and hence (i) follows. Item (ii) then follows immediately. Now substitute (17) back into the objective, using p j (v) ξ j (v) = 0 , to find that it reduces to

Λ = Eγ (V) = E log j H j e ṽ (29)
We may then use (29) to determine p 0 . Now apply Eq. ( 6) to establish part (iii) of the proposition.

Proof of proposition 4. Assume, towards a contradiction, that p 0 a > 0. Then

p 0 a = E    H a e Vc S c p 0 N c=1 b H b {e Vc S c (p 0 )} N c=1    (30) < E    H a e Va S c p 0 N c=1 b H b {e Va S c (p 0 )} N c=1    (31) = E    e Va H a S c p 0 N c=1 e Va b H b {S c (p 0 )} N c=1    = E   p 0 a b p 0 b   = p 0 a . (32)
The first inequality (31) follows from cyclic monotonicity, which is a property of the gradient of convex functions. (See, for instance, Rockafellar (1970, Thm. 23.5).) Since the surplus function W is convex, its gradient, corresponding to the choice probabilities p(•) is a cyclic monotone mapping, implying that p e va S c p 0 N c=1 -p e vc S c p 0 N c=1

• e va S c p 0 N c=1 -e vc S c p 0 N c=1 ≥ 0.

All the terms within the second pair of brackets on the LHS are ≤ 0, except for the a-th term, which is equal to zero. In order to satisfy the inequality, then, we must have p a e va S c p 0 N c=1 ≥ p i e vc S c p 0 N c=1 with the inequality strict with positive probability. Otherwise, with the final inequality strict with positive probability. Hence, we conclude that p 0 a = 0.

In the case of the Shannon entropy, Corollary 4 can be strengthened considerably. In that case, any alternative that is dominated by another alternative in all states of the world will never be chosen, as shown in the following corollary:

Corollary 7 Let S be the identity. Suppose that option a is dominated by option d in the sense that ∀v ∈ V : v a ≤ v d with strict inequality for some v. Then p 0 a = 0.

Proof. Suppose to get a contradiction that p 0 a > 0. From ( 13), obtain that for all options a

1 = p 0 a p 0 a = 1 p 0 a Ep a (V) = E   exp (V a ) b exp (V b ) p 0 b   . Then E   exp (V d ) b exp (V b ) p 0 b   > 1,
which is a contradiction.

Proof of Proposition 6. Independence: By independence, we have, for all i, p i (v) = k i , a constant. Then p 0 i = k i and κ S (p(•), µ) = 0. Convexity: Consider two sets of choice probabilities p 1 (v) , p 2 (v) , v ∈ V, where both have the same implied unconditional probabilities Ep 1 (V) = Ep 2 (V). For ρ ∈ [0, 1], define p ρ as the convexification ρp 1 (v) + (1 -ρ) p 2 (v). Lemma 8 H is invertible.

Proof of Lemma 8. We shall make use of Ruzhansky and Sugimoto's 2015 invertibility result applied to H. The Jacobian of ṽ → H e ṽ is e W (ṽ) ∂W (ṽ) ∂v i ∂W (ṽ) ∂v j + e W (ṽ) ∂ 2 W (ṽ) ∂v i ∂v j . The first matrix is positive definite since all choice probabilities are positive, the second matrix is positive semidefinite due to the convexity of W , hence this matrix is everywhere positive definite and then the Jacobian determinant of ṽ → H e ṽ never vanishes. This implies in turn that the Jacobian determinant of the composition y → log y → H (y) never vanishes. It remains to show that inf y∈∆ H (y) > 0. But y ∈ ∆ implies that H (y) = e W (log y) ∇W (log y)

≥ e E max j {log y j +ε j } J -1/2 ≥ e max j {log y j +Eε j } J -1/2 = max j y j e Eε j J -1/2 ≥ y 1 e Eε 1 , ..., y J e Eε N J -1

≥   N j=1 e -2Eε j   -1 J -1 > 0,
where we first used that ∇W is on the unit simplex, second that the max operation is convex, third that the sup-norm bounds the euclidean norm, and fourth that the minimum of y 1 e Eε 1 , ..., y N e Eε N on the unit simplex is attained at

y j = e -2Eε j N k=1
e -2Eε k -1

, j = 1, ..., N .

B Example: Consideration sets and failure of regularity

Next, we consider a fully solved out example illustrating the possibility of zero unconditional choice probabilities and failure of regularity, which can occur in the rational inattention framework but not in the discrete choice model, and represent an important point of difference between the two models. Matejka and McKay (2015, pp. 293ff) have demonstrated that failures of regularity can occur in the RI model under Shannon entropy. We show that such failures also occur in a GERI model, in particular for the nested logit information cost function introduced in Section 4 of the main text. Consider a setting with four choice options. Table 2 lists the valuation vectors for these four options in the three equiprobable states of the world. We consider both the Shannon and GERI-nested logit models. (For the nested logit specification, we assume that nest 1 consists of choices (1,2) with nesting parameter ζ 1 = 0.7, and nest 2 consists of choices (3,4) with parameter ζ 2 = 0.8.

For each model, we compute the optimal unconditional probabilities (which as in the previous example, requires solving the fixed-point equation ( 16)) first for the choice set {1, 2, 3}, and then for the expanded choice set {1, 2, 3, 4}. This example illustrates how adding option 4 to the choice set can results in increases in the choice probabilities of choices (1,2,3) thus showing a failure of the regularity property. The optimal unconditional probabilities are shown in Table 3. Qualitatively the results are the same between both the Shannon and GERI-nested logit specifications. With the smaller set of options, we see that only options 1,2 are

  i =a p i e va S c p 0 N c=1 -p i e vc S c p 0 N c=1 > 0 and p e va S c p 0 N c=1 -p e vc S c p 0 N c=1 • e va S c p 0 N c=1 -e vc S c p 0 N c=1 = c =a p c e va S c p 0 N c=1 -p c e vc S c p 0 N c=1 [e va -e vc ] S c p 0 ≤ max c =a (e va -e vc ) S c p 0 c =a p c e va S c p 0 N c=1 -p c e vc S c p 0 N c=1 ≤ 0

  Then we would like to show thatρκ S (p 1 (•), µ) + (1 -ρ) κ S (p 2 (•), µ) ≥ κ (p ρ (•), µ) . But ρκ S (p 1 (•), µ) + (1 -ρ) κ S (p 2 (•) µ) -κ (p ρ (•), µ) = -ρΩ S (p 1 ) -(1 -ρ) Ω S (p 2 ) + Ω S (ρp 1 + (1 -ρ) p 1 ) ,which is positive by concavity of Ω S (p) (Proposition 2(ii)).

  ζg i j∈g i e ṽj /ζg i

	•	e	ζg i log	j∈g i	e ṽj /ζg i	(22)
		all nests g e	ζg log

j∈g e ṽj /ζg .

Table 1 :

 1 Choice Probabilities in GERI model: Nested Logit vs. Multinomial Logit 1,2 and 3, which constitute the larger nest, and smaller for options 4,5 which constitute the smaller nest. (However, within nest, the choice probabilities are identical.) The non-uniform choice probabilities for the nested logit model reflect the cost spillovers across options in the structure of the nested logit information cost function.

	1

Throughout this paper, we will use the terms "additive random utility model" and "discrete choice model" interchangeably.

This complements work by[START_REF] Hébert | Rational Inattention with Sequential Information Sampling[END_REF], who also consider generalizations of the information cost function.

The convexity of W (•) follows from the convexity of the max function. Differentiability follows from the absolute continuity of .

By direct differentiation of (3), and applying the Williams-Daly-Zachary theorem, we have qi(ṽ) = Hi(e W (ṽ) )/e W (ṽ) for all i. Imposing the summability restriction i qi(ṽ) = 1 we have i Hi(e W (ṽ) ) = e W (ṽ) leading to Eq. (4).

For a convex function g(x), its convex conjugate function is defined as g * (y) = maxx {x • y -g(x)}, which is also convex. Roughly speaking, the gradients (or sub-gradients, in case of non-differentiability) of g(x) and g * (y) are inverse mappings to each other.

See Chiong, Galichon, and Shum (2016). Additionally, we conjecture that log Si(q) = -E[ i|ui ≥ uj, j = i] for i = 1, . . . , N . For the multinomial logit case, corresponding to S(q) = q,[START_REF] Mcfadden | Modelling the choice of residential location[END_REF] showed that γ -log qi = E[ i|ui ≥ uj, j = i], for γ being Euler's constant.

In fact, the necessity part of Assumption 2 arises immediately from the results in this section. As ṽi → -∞, qi(ṽ) → 0, which by (4) implies that Hi e ṽ → 0. Then, since log S(q(ṽ)) = ṽ -log j Hj(e ṽ ), we have log S1(q) → -∞ (by homogeneity of H, we may suppose that log j Hj(e ṽ ) is a constant).

To see this, consider the solution to the GERI problem given in Eq. (17) and define ṽ = v + log S(p 0 ). Let p 0 i = 0. Then by assumption 2 it follows that log Si(p 0 ) = -∞, or equivalently, ṽi -→ -∞ and hence pi(v) = 0 for all v ∈ V.

Because of the possibility of zero choice probabilities for some options, GERI models can also generate failures of the "regularity" property (adding an option to a choice set cannot increase the choice probability for any of the original choices). See section B in the Appendix for an example.

See, for instance,Maddala (1983, Chap. 2), and[START_REF] Anderson | Discrete Choice Theory of Product Differentiation[END_REF].

In the nested logit case, we obtained the unconditional distribution by iterating over the fixed point relation p 0 = Ep(V), starting from the multinomial logit distribution.

In a similar vein, Webb (2016) demonstrates an equivalence between random utility models and bounded-accumulation or drift-diffusion models of choice and reaction times used in the neuroeconomics and psychology literature.

See, for instance,[START_REF] Berry | Automobile Prices in Market Equilibrium[END_REF],[START_REF] Mcfadden | Mixed MNL Models for Discrete Response[END_REF],[START_REF] Fox | The random coefficients logit model is identified[END_REF].

As an example, the function g(p) = -N i=1 log(pi) is not a generalized entropy function; thus a rational inattention model using this as an information cost function would lie outside the GERI framework.

State: chosen with positive probabilities. When option 4 is added, however, then option 1 drops out of the consideration set, and only options 3,4 are chosen with positive probability. This demonstrates a failure of regularity, as the addition of choice 4 increases the prior choice probability for choice 1. (Moreover, note that with the expanded choice set, option 2 is chosen with zero probability, even though it is not inferior in all states of the world, which demonstrates that the characterization of consideration sets in Corollary 4 is not exhaustive.) Basically, the addition of choice 4 allows agents to form an effective "hedge" in conjunction with choice 3. In the state when choice 3 yields a low payoff (state v 2 ), choice 4 yields a high payoff; on the contrary, when choice 4 yields a lower payoffs (states v 1 and v 3 ), choice 3 yields high payoffs.