Rémi Maurice

A polynomial realization of the Hopf algebra of uniform block permutations

We investigate the combinatorial Hopf algebra based on uniform block permutations and we realize this algebra in terms of noncommutative polynomials in infinitely many bi-letters. Résumé. Nous étudions l'algèbre de Hopf combinatoire dont les bases sont indexées par les permutations de blocs uniformes et nous réalisons cette algèbre en termes de polynômes non-commutatifs en une infinité de bi-lettres.

combinatorial Hopf algebras, uniform block permutations

1 Introduction.

For several years, Hopf algebras based on combinatorial objects have been thoroughly investigated. Examples include the Malvenuto-Reutenauer Hopf algebra FQSym whose bases are indexed by permutations [START_REF] Malvenuto | Duality between quasi-symmetric functions and the Solomon descent algebra[END_REF][START_REF] Duchamp | Noncommutative symmetric functions VI: Free quasisymmetric functions and related algebras[END_REF][START_REF] Aguiar | Structure of the Malvenuto-Reutenauer Hopf algebra of permutations[END_REF], the Loday-Ronco Hopf algebra PBT whose bases are indexed by planar binary trees [START_REF] Loday | Hopf algebra of the planar binary trees[END_REF], the Hopf algebra of free symmetric functions FSym whose bases are indexed by standard Young tableaux [START_REF] Poirier | Algèbres de Hopf de tableaux[END_REF].

The product and the coproduct, which define the structure of a combinatorial Hopf algebra, are, in general, rules of composition and decomposition given by combinatorial algorithms such as the shuffle and the deconcatenation of permutations, as in FQSym [START_REF] Duchamp | Noncommutative symmetric functions VI: Free quasisymmetric functions and related algebras[END_REF], of packed words, as in WQSym [START_REF] Novelli | Polynomial realizations of some trialgebras[END_REF] or of parking functions, as in PQSym [START_REF] Novelli | Hopf algebras and dendriform structures arising from parking functions[END_REF]. These can also be given by the disjoint union and the admissible cuts of rooted trees, as in the Connes-Kreimer Hopf algebra [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF], or by concatenation and subgraphs/contractions of graphs [START_REF] Manchon | On bialgebras and Hopf algebras of oriented graphs[END_REF].

Computing with these structures can be difficult, and polynomial realization can bring up important simplifications.

We can see a polynomial realization as the image by an injective Hopf algebra morphism from a combinatorial Hopf algebra to an algebra of polynomials (commutative or not).

The idea is to encode the combinatorial objects by polynomials, in such a way that the product of the combinatorial Hopf algebra become the ordinary product of polynomials, and that the coproduct be given by the disjoint union of alphabets, endowed with some extra structure such as an order relation. This can be done, e.g., for FQSym [START_REF] Duchamp | Noncommutative symmetric functions VI: Free quasisymmetric functions and related algebras[END_REF], WQSym [START_REF] Novelli | Polynomial realizations of some trialgebras[END_REF], PQSym [START_REF] Novelli | Hopf algebras and dendriform structures arising from parking functions[END_REF] and Connes-Kreimer [START_REF] Foissy | Polynomial realizations of some combinatorial Hopf algebras[END_REF].

In this paper, we realize the Hopf algebra UBP of uniform block permutations introduced by Aguiar and Orellana [START_REF] Aguiar | The Hopf algebra of uniform block permutations[END_REF]. We begin by recalling the preliminary notions of uniform block permutations in Section 2. We recall, in Section 3, the Hopf algebra structure of uniform block permutations, we translate this subm. to DMTCS c by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France structure via the decomposition and we describe the dual structure. Using polynomial realizations of Hopf algebras based on permutations and set partitions, we realize this algebra in Section 4. We finish by finding the Hopf algebra WQSym as a quotient of UBP and the dual as a subalgebra of UBP * . Definition 1 A uniform block permutation of size n is a bijection between two set partitions of {1, • • • , n} that maps each block to a block of the same cardinality.

Notation 1 Let A = {A 1 , A 2 , • • • , A k } and A ′ = {A ′ 1 , A ′ 2 , • • • , A ′ k }
be two set partitions of the same size. We denote the uniform block permutation f : A -→ A ′ for which the image of A i is A ′ i by

A 1 A 2 • • • A k A ′ 1 A ′ 2 • • • A ′ k .
Example 1 Here are all uniform block permutations of size 1 and 2:

n = 1 1 1 n = 2 12 12 , 1 2 1 2 , 1 2 2 1 .
In the case where the set partitions consist of singletons (i.e., A = {1}, • • • , {n} ), we find the permutations of size n.

1. C is a union of blocks of A 2. f (C) is a union of blocks B i of B.
The image of C is the union of images g(B i ).

Note that if f is a permutation, then the set partition C ′ is B ′ , and if g is a permutation, then the set partition C is A. The data of a permutation and a set partition is enough to describe a uniform block permutation. Given a uniform block permutation f : A -→ A ′ of size n, there is a permutation σ in S n such that

A ′ = H∈A x∈H {σ(x)} .
Conversely, a set partition of the set {1, • • • , n} and a permutation of S n determine a uniform block permutation of size n. Any uniform block permutation can thus decompose (non-uniquely) as

f = σ • Id A = Id A ′ • σ. (1)
The first equality expresses the fact that one can group the images σ i whose indices are in the same block in A, the second expresses the fact that one can group the images σ i whose values are in the same block in A ′ .

3 Algebraic structures.

In [START_REF] Aguiar | The Hopf algebra of uniform block permutations[END_REF], Aguiar and Orellana introduced a Hopf algebra whose basis is the set of uniform block permutations. It will be denoted by UBP, standing for Uniform Block Permutations. Let us recall the convolution of two permutations. Let τ and τ ′ be two permutations:

τ * τ ′ = σ=u•v std(u)=τ std(v)=τ ′ σ
where the standardization is a process that associates a permutation with a word. The standardized of w is defined as the permutation having the same inversions as w. For example: std(a 2 a 3 a 2 a 1 a 6 a 1 a 4 ) = 3541726, 12 * 21 = 1243 + 1342 + 1432 + 2341 + 2431 + 3421.

The product and the coproduct are defined as follows.

Definition 4 Let f and g be two uniform block permutations of respective sizes m and n. The convolution, denoted by f * g, is the sum of all uniform block permutations of the form

1 2 • • • m + n σ 1 σ 2 • • • σ m+n • (f × g).
where

std(σ 1 • • • σ m ) = 12 • • • m and std(σ m+1 • • • σ m+n ) = 12 • • • n (i.e., σ appears in 12 • • • m * 12 • • • n).
Example 4

1 1 * 12 12 = 1 2 3 1 2 3 + 1 2 3 2 1 3 + 1 2 3 3 1 2 • 1 23 1 23 = 1 23 1 23 + 1 23 2 13 + 1 23 3 12 .
We denote by std(E) the standardization of a collection of disjoint sets of integers obtained by numbering each integer a in E by one plus the number of integers in E smaller than a.

Example 5

std {1, 6}, {9, 13}, {3, 5, 12} = {1, 4}, {5, 7}, {2, 3, 6} .

We set

C |{1,••• ,i} = H ∈ C; H ⊂ {1, 2, • • • , i} , C |{i+1,••• ,n} = std C -C |{1,••• ,i}
and let I(C) be the set of integers i such that

{1, 2, • • • , n} = C |{1,••• ,i} × C |{i+1,••• ,n}
for any set partition C.

Example 6 Let C = {1, 3}, {2}, {4, 6, 7}, {5} be a set partition:

C {1,••• ,2} = {2} C {1,••• ,3} = {1, 3}, {2} C {4,••• ,7} = {1, 3, 4}, {2} I(C) = {0, 3, 7} Definition 5 Let f : A -→ A ′ be a uniform block permutation of size n. By Lemma (2.2) in [1], i is in I(A ′
) if and only if there exists a unique permutation σ in S n appearing in the convolution product

12 • • • i * 12 • • • n -i and unique uniform block permutations f (i) of size i and f ′ (n-i) of size n -i such that f = (f (i) × f ′ (n-i) ) • 1 2 • • • n (σ -1 ) 1 (σ -1 ) 2 • • • (σ -1 ) n .
The coproduct is then

∆(f ) := i∈I(A ′ ) f (i) ⊗ f ′ (n-i) .
Example 7 One can rewrite this product and coproduct using the decomposition of a uniform block permutation. Given f : A -→ A ′ and g : B -→ B ′ two uniform block permutations,

∆
f = σ • Id A = Id A ′ • σ and g = τ • Id B = Id B ′ • τ , we have f * g =(σ * τ ) • Id A×B (2)
∆(f ) = i∈I(A ′ ) f |{1,••• ,i} ⊗ f |{i+1,••• ,n} (3)
where It is proved in [START_REF] Aguiar | The Hopf algebra of uniform block permutations[END_REF] that UBP is a self-dual Hopf algebra. We can describe the dual structure. Let us first define another product and coproduct of uniform block permutations. Let f : A -→ A ′ and g : B -→ B ′ be two uniform block permutations,

f |{1,••• ,i} =Id A ′ |{1,••• ,i} • σ |{1,••• ,i} f |{i+1,••• ,n} =Id A ′ |{i+1,••• ,n} • std(σ |{i+1,••• ,n}
f = σ • Id A = Id A ′ • σ and g = τ • Id B = Id B ′ • τ : f ✂ g =Id A ′ ×B ′ • (σ ✂ τ ) (4) ∆ 1 (f ) = i∈I(A) f 1 • • • f i ⊗ f i+1 • • • f n (5)
where

f 1 • • • f i =std(σ 1 • • • σ i ) • Id A |{1,••• ,i} f i+1 • • • f n =std(σ i+1 • • • σ n ) • Id A |{i+1,••• ,n} .
Example 10 The two structures of Hopf algebra are dual to each other for the scalar product

1 1 ✂ 12 12 =Id {{1},{2,3}}
f, g = δ f,g -1 . (6)
In other words, we have:

∆f, g ⊗ h = f, g ✂ h (7) ∆ 1 f, g ⊗ h = f, g * h . (8)
4 Polynomial realization.

Given two alphabets B = {b i ; 0 ≤ i} and C = {c i ; 0 ≤ i}, with C totally ordered, we consider an alphabet A = b i c j ; 0 ≤ i, j of bi-letters endowed with two relations:

• for all i and k, b i c j b k c l if and only if c j ≤ c l ,
• for all j and l, b

i c j ≡ b k c l if and only if b i = b k .
The product of two biwords is the concatenation

u 1 v 1 u 2 v 2 = u 1 u 2 v 1 v 2 .
We extend the definition of the standardization to biwords. Let w = u v be a biword. The standardized of w is the permutation std(v).

Example 11

std b 1 b 1 b 3 b 1 b 2 c 2 c 3 c 2 c 1 c 6 = 24315.
Let f : A -→ A ′ be a uniform block permutation. Let ξ f be the minimum permutation (for the lexicographic order) such that f = ξ f • Id A . We say that a biword w is f -compatible if std(w) = ξ f and -if i and j are in the same block in A, then the bi-letters w i and w j satisfy w i ≡ w j .

We set:

G f (A) = w;w is f -compatible w. ( 9
)
Example 12 Let f = 13 256 4 15 234 6 be a uniform block permutation. The permutation σ = 125634

is the minimum permutation associated with f and we have

G f (A) = i,j,k std(c1c2•••c6)=σ b i b j b i b k b j b j c 1 c 2 c 3 c 4 c 5 c 6 .
Theorem 1 The polynomials G f (A) provide a realization of the Hopf algebra of uniform block permutations. That is to say

• G f (A) G g (A) = h∈f * g G h (A)
• Let A ′ be an alphabet isomorphic to the alphabet A such that, for any bi-letter w ′ of A ′ and for any bi-letter w of A, we have w w ′ .

Define A + A ′ as the disjoint union of A and A ′ endowed with the relations and ≡. If we allow the bi-letters of A and A ′ to commute and identify P (A)Q(A ′ ) with P ⊗ Q, we have:

G f (A + A ′ ) = ∆(G f ). ( 10
)
Example 13

G   1 1   (A)G   1 1   (A) = b∈B c∈C b c 2 = b1,b2∈B c1,c2∈C b 1 b 2 c 1 c 2 = b1,b2∈B c1,c2∈C;c1≤c2 b 1 b 2 c 1 c 2 + b1,b2∈B c1,c2∈C;c1>c2 b 1 b 2 c 1 c 2 =G   1 2 1 2   (A) + G   1 2 2 1   (A). G   12 12   (A)G   1 1   (A) = b∈B std(v)=12 bb v b∈B c∈C b c = i1,i2 std(v1)=12 std(v2)=1 b i1 b i1 b i2 v 1 v 2 = i1,i2 c1≤c2≤c3 b i1 b i1 b i2 c 1 c 2 c 3 + i1,i2 c1≤c3<c2 b i1 b i1 b i2 c 1 c 2 c 3 + i1,i2 c3<c1≤c2 b i1 b i1 b i2 c 1 c 2 c 3 =G   12 3 12 3   + G   12 3 13 2   + G   12 3 23 1   .
Example 14 We set τ = 2431.

G   13 24 13 24   (A + A ′ ) = i,j c1≤c2≤c3≤c4 b i b j b i b j c 1 c 2 c 3 c 4 + i,j c ′ 1 ≤c ′ 2 ≤c ′ 3 ≤c ′ 4 b ′ i b ′ j b ′ i b ′ j c ′ 1 c ′ 2 c ′ 3 c ′ 4 =G   13 24 13 24   (A) + G   13 24 13 24   (A ′ ). G   13 2 4 23 4 1   (A + A ′ ) = i,j,k std(c1c2c3c4)=τ b i b j b i b k c 1 c 2 c 3 c 4 + i,j,k std(c ′ 1 c ′ 2 c ′ 3 c4)=τ b ′ i b ′ j b ′ i b k c ′ 1 c ′ 2 c ′ 3 c 4 + i,j,k std(c1c ′ 2 c3c4)=τ b i b ′ j b i b k c 1 c ′ 2 c 3 c 4 + i,j,k std(c ′ 1 c ′ 2 c ′ 3 c ′ 4 )=τ b ′ i b ′ j b ′ i b ′ k c ′ 1 c ′ 2 c ′ 3 c ′ 4 =G   13 2 4 23 4 1   (A) + G   1 1   (A)G   13 2 12 3   (A ′ ) + G   12 3 23 1   (A)G   1 1   (A ′ ) + G   13 2 4 23 4 1   (A ′ ).
5 Packed words.

We recall the definition of a packed word and the structure of Hopf algebra given in [START_REF] Novelli | Polynomial realizations of some trialgebras[END_REF]. The packed word pack(w) associated with a word w ∈ A * is the word u obtained by the following process. If b 1 < b 2 < • • • < b r are the letters occuring in w, u is the image of w by the homomorphism b i -→ a i . A word w is said to be packed if pack(w) is w.

5.1 WQSym * .

The Hopf algebra WQSym * is the K-vector space generated by packed words endowed with the shifted shuffle product and the coproduct given by, using notations in [START_REF] Foissy | Polynomial realizations of some combinatorial Hopf algebras[END_REF]:

u ✂ v =u ✁ v[max (u)], (11)
∆(w) = u•v=w pack(u) ⊗ pack(v). (12)
Let Ψ be the map that associates the word w of the length n with a permutation σ of size n and a set partition A of {1, 2, • • • , n} constructed by the following process. The ith letter u i of u is the minimum of the letters σ j whose values σ j are in the same block in A as σ i . The word w is obtained by packing. The map Ψ induces a linear map from the vector space generated by uniform block permutations to that generated by packed words. We define it by writing uniform block permutations f in the form Id A ′ • σ and applying Ψ to σ and A ′ . We also call this map Ψ. If we endow the vector spaces with the shifted shuffle product, the map Ψ is an algebra morphism.

Proposition 1 Let f and g be two uniform block permutations.

Ψ(f ✂ g) = Ψ(f ) ✂ Ψ(g). (13)
✂ Ψ 1 2 1 2 .
The map Ψ cannot be a coalgebra morphism as shown in the example

Ψ ⊗ Ψ • ∆ 12 3 23 1 =Ψ ⊗ Ψ ⊗ 12 3 23 1 + 12 12 ⊗ 1 1 + Ψ ⊗ Ψ 12 3 23 1 ⊗ =ε ⊗ 221 + 11 ⊗ 1 + 221 ⊗ ε =ε ⊗ 221 + 1 ⊗ 21 + 11 ⊗ 1 + 221 ⊗ ε =∆(221).
The idea is that the coproduct on packed words deconcatenates the word in all suffixes and prefixes, whereas the coproduct on uniform block permutations preserves the blocks. This suggests to define an another coproduct:

∆ ′ (w) = u•v=w alph(u)∩alph(v)=∅ pack(u) ⊗ pack(v).
Endowed with the shifted shuffle product ✂ and with the coproduct ∆ ′ , the vector space generated by packed words is a combinatorial Hopf algebra denoted by WQSym * '.

Proposition 2

The Hopf algebras WQSym * and WQSym * ' are isomorphic.

The morphism of Hopf algebras is given by the following process. We define the following relation on packed words of the same length. Let u and v be two packed words, the cover relation is u v if and only if u is an isomorphism of Hopf algebras. For example, the image of a permutation is a permutation, the image of the word of length n consisting only of 1 is the sum of all the non-decreasing packed words of length n.

• std(u) = std(v) • there exists a i such that u k = v k if v k ≤ i v k -1 otherwise.
✂ 1 + 12 ✂ 1 =Γ(11) ✂ Γ(1) Γ ⊗ Γ(∆(121)) =121 ⊗ ε + 132 ⊗ ε + 12 ⊗ 1 + 1 ⊗ 21 + ε ⊗ 121 + ε ⊗ 132 =∆ ′ (121) + ∆ ′ (132)
=∆ ′ (Γ(121)).

Proposition 3

The map Ψ is a surjective Hopf algebra morphism from UBP * to WQSym * '.

WQSym.

We denote by WQSym the K-vector space generated by the packed words endowed with the convolution product and the coproduct given by

u * W v = w=u ′ •v ′ ∈P W ;pack(u ′ )=u,pack(v ′ )=v w, (14)
∆(u) = 0≤k≤max (u) u |{1,••• ,k} ⊗ pack(u |{k+1,••• ,max (u)} ) (15)
The dual WQSym' of WQSym * ' is the Hopf algebra endowed with the product

u * ′ W v = w=u ′ •v ′ ∈P W pack(u ′ )=u pack(v ′ )=v alph(u)∩alph(v)=∅ w ( 16
)
for any packed word u and v, and the coproduct ∆.

Proposition 4

The Hopf algebra WQSym' and WQSym are isomorphic.

In fact, the map Γ induces a dual map. This is

Γ * : WQSym' -→ WQSym w -→ u w u.

Proposition 5

 5 For example: Γ * (1212334) = 1212334 + 1212223 + 1212333 + 1212222.This map is an isomorphism of Hopf algebras. Example 18 Γ * (212 * ′ W 11) =Γ * (21233 + 31322 + 32311) =(21233 + 21222) + (31322 + 21211) + 32311 =212 * W 11 =Γ * (212) * W Γ * (11) Γ * ⊗ Γ * • ∆(312341) =Γ * ⊗ Γ * ε ⊗ 312341 + 11 ⊗ 2123 + 121 ⊗ 112 + 31231 ⊗ 1 + 312341 ⊗ ε =ε ⊗ (312341 + 312331) + 11 ⊗ (2123 + 2122)+ 121 ⊗ (112 + 111) + 31231 ⊗ 1 + (312341 + 312331) ⊗ ε =∆(312341 + 312331) =∆ Γ * (312341) The map ϕ : WQSym ′ -→ UBP w -→ f ;Ψ(f )=w f is an injective morphism of Hopf algebras.

  Let f : A -→ A ′ and g : B -→ B ′ be two uniform block permutations of the same size n. The composition g • f : C -→ C ′ of f and g is defined by the following process. The blocks C of the set partition C are the subsets of {1, • • • , n} which are minimal for the two properties

	Definition 2 Let f and g be two uniform block permutations. The concatenation of f and g, denoted
	by f × g, is the uniform block permutation obtained by adding the size of f to all entries of g and
	concatenating the result to f .
	Example 2
	13 2 23 1	×	13 2 46 5 46 2 15 3	=	13 2 46 5 79 8 23 1 79 5 48 6	.
	Definition 3

  ).

	Example 8
			1 1	*	12 12	=1 • Id {1}	* 12 • Id {1,2}
						=(1 * 12) • Id {1},{2,3}
						=(123 + 213 + 312) • Id {1},{2,3}
						=	1 23 1 23	+	1 23 2 13	+	1 23 3 12	.
	∆	13 256 4 15 234 6	=∆(Id A • 125634)
			=ε ⊗ Id A • 125634 + Id {1,5},{2,3,4}	• 12534 ⊗ Id {1}	• 1 + Id A • 125634 ⊗ ε
			=	⊗		13 256 4 15 234 6	+	13 245 15 234	⊗	1 1	+	13 256 4 15 234 6	⊗	.

Example 9

We set A = {1, 5}, {2, 3, 4}, {6} .

Acknowledgements

The author would like to thank Jean-Yves Thibon for his advice and reading of this paper.

2 Definitions and decompositions.


Jana Trgalová

Ana Paula Jahn

THE IMPACT OF THE INVOLVEMENT OF TEACHERS IN A RESEARCH ON RESOURCE QUALITY ON THEIR PRACTICES

Since four years, a group of seven secondary school mathematics teachers and teacher educators has been involved in a research project dealing with the issue of dynamic geometry resource quality. The aim of this paper is to examine the impact of this involvement on their practices both as teachers and teacher educators. Based on the analysis of various resources produced by the group members before their participation in the research and nowadays, as well as on the group's auto-analysis of the evolution of their own practices, we could highlight a significant evolution in their way of using dynamic geometry in a classroom, as well as in their teacher training offer and content.

Research in mathematics education we carry out relies mostly on participatory methodologies, such as design-based research [START_REF] Dbrc | Design-Based Research: An Emerging Paradigm for Educational Inquiry[END_REF], which involve researchers and practicing teachers who are not mere experimental subjects, but they take an active part in all phases of the research. Like in other research projects, the focus is on the research itself and its outcomes and the issue of the teachers' professional development fostered by their participation to the research is not raised.

The question how the teachers' involvement into a research contributes to their professional development is central (Burns 2010). We report about an experience of a research group composed of three mathematics education researchers and seven secondary school math teachers (called DG-group in the sequel) working on dynamic geometry (DG) resource quality issues. We attempt to highlight the impact of the teachers' involvement into this research on their practices.

The paper is organised as follows. First, we briefly describe the research project on DG resource quality that gathered together researchers and teachers. This project will be referred to as I2G project. Next, we present the theoretical framework and the methodology we used to examine the impact of the teachers' involvement in the I2G project on their practices. Finally, we discuss the most significant findings and propose some concluding remarks.

I2G PROJECT

The I2G project, which ran between September 2008 and June 2012, was conducted in the framework of the Intergeo European project [1], aiming at developing a community of DG users all over Europe around an open web-based repository [2] specifically designed for sharing resources and practices related to the use of DG in mathematics teaching. In order to help platform users identify suitable resources regarding their instructional aim and context of use, as well as to allow the available resources to be improved, two main tools have been developed and implemented into the repository: a search engine based on mathematical notions and competencies ontology to help searching for relevant resources, and a resource quality review questionnaire helping the users analyse available resources and highlight aspects to be improved. Three mathematics education researchers and the DG-group were in charge of designing and testing the questionnaire. In the next section, we present the questionnaire and its design process, focusing on the roles of the researchers and the teachers involved in the I2G project.

Design of the questionnaire for DG resource analysis

The questionnaire, which is the main tool for the resource quality assessment in the repository (Fig. 1), was designed in a cyclical process consisting in the elaboration of its successive versions, followed by their tests and subsequent improvements. This methodology can be considered as a design-based research, in which "development and research take place through continuous cycles of design, enactment, analysis, and redesign" (DBRC 2003, p. 5), blending theory-driven design with empirical research. The first version of the questionnaire was designed by the researchers drawing on research results related to the use of DG in math teaching and learning. It proposed eight general questions related to the dimensions of a DG resource considered as critical with respect to its quality, such as technical aspect, mathematical content validity, instrumental aspect, and didactical and pedagogical implementation. Later, a ninth dimension related to the resource ergonomics has been added. Each of these questions can be developed into a set of more detailed criteria related to the corresponding dimension (Fig. 1). The theoretical considerations underpinning the choice of the dimensions and the definition of the criteria are exposed in some details in [START_REF] Trgalová | Quality assessment process for dynamic geometry resources in Intergeo project: rationale and experiments[END_REF].

In order to make the questionnaire accessible to and usable by teachers, its elaboration has been done in a close collaboration with seven secondary mathematics teachers (DG-group), according to the schema in Fig. 2. Each version of the questionnaire was first reviewed by the DG-group who provided suggestions related to the relevance and the clarity of the quality criteria. An improved version was then tested with teachers in various contexts, such as pre-service or in-service teacher training programs or workshops (Jahn et al. 2009[START_REF] Trgalová | Quality assessment process for dynamic geometry resources in Intergeo project: rationale and experiments[END_REF][START_REF] Trgalová | Analyse de ressources comme moyen de développement professionnel des enseignants[END_REF]. The questionnaire was then re-designed to take into account the outcomes of the tests. As it can be seen in the Figure 2, the teachers from the DG-group played an important role in the I2G research project and contributed to it significantly. During the last year of the project, we became naturally interested in what was the benefit of the teachers' involvement in the project in terms of their professional development. Below we outline the methodology and the theoretical background of this study

METHODOLOGY

The teachers from the DG-group started collaborating in 1996 within the Institute for Research in Mathematics Teaching in Lyon [3] as a group of "users and designers of DG resources" [START_REF] Bourgeat | De l'utilisation à la formation : géométrie dynamique et logiciels mathématiques[END_REF]. In 2008, the group volunteered to join the I2G project and it collaborated until recently with math education researchers on the DG resource quality issues. Every year since 2003, the group offers training courses aiming at helping other math teachers master and integrate DG systems into their practices. All group members prepare together their courses during their regular meetings and produce various resources (documents, tools…), although only two or three of them, in turn, are in charge each time of the implementation of the courses.

For lack of possibility to set up a long-term observation of classroom and teacher training practices of the DG-group members, we gathered three kinds of data in order to identify possible evolutions of their practices. First we collected teacher training resources the group has produced since 2003. Second, recently the group was asked to reflect on changes in their own practices that the group members could observe since their involvement in the I2G project. This introspective activity, conducted during an informal interview with the teachers and summarized in the form of a workshop communication [START_REF] Bourgeat | De l'utilisation à la formation : géométrie dynamique et logiciels mathématiques[END_REF], yielded many interesting observations related mostly to the teaching practices, some of which are reported in this paper. Finally, we have analyzed some of the teachers' reviews and comments of resources in the Intergeo platform, which can also shed light on their practices with using DG in math teaching.

THEORETICAL LENS USED TO ANALYSE THE DATA

In the collected data we wished to observe indicators of the evolution of the teachers' practices with using DG in their teaching math or teacher training, as well as indicators of the changes in their sensitivity to teacher training issues. This concern led us to choose the following theoretical frameworks.

Instrumental approach and double instrumental genesis

Numerous research studies on the information and communication technologies (ICT) integration adopt the instrumental approach [START_REF] Rabardel | People and technology -a cognitive approach to contemporary instruments[END_REF] as a theoretical framework specifically designed for studying teaching and learning phenomena involving technology. The instrumental approach relies on a distinction between an artefact, a tool available to an individual, and an instrument, which is the result of a process of appropriation of the tool by the individual when s/he uses it in order to achieve a given task. The process of transforming an artefact into an instrument is called instrumental genesis. Some of these studies stress the complexity of technology integration, which requires a double instrumental genesis in teachers: a first genesis of an instrument for achieving mathematics tasks, and a second one of an instrument for achieving educational tasks [START_REF] Acosta | Implications dans la formation d'enseignants à l'utilisation de Cabri en classe de géométrie[END_REF]. [START_REF] Haspekian | The co-construction of a mathematical and a didactical instrument[END_REF] evokes a personal genesis transforming a given tool into a mathematical instrument, and a professional genesis transforming it into a didactical instrument. According to [START_REF] Trouche | Managing the complexity of human/machine interactions in computerized learning environments: guiding students' command process through instrumental orchestrations[END_REF], the ICT integration requires from the teacher to be aware of the potentialities and constraints of artefacts, which is necessary to design suitable mathematical tasks. Moreover, the teacher has to be able to implement these tasks into the classroom and to foresee the spatial and temporal classroom management.

The author introduces the term instrumental orchestration to refer to the didactical management of the artefact in a classroom. [START_REF] Drijvers | Instrumental orchestration: theory and practice[END_REF] define the instrumental orchestration as "the intentional and systematic organisation and use of the various artefacts available in [a] computerised learning environment by the teacher in a given mathematical task situation, in order to guide students' instrumental genesis" (p. 1350).

These considerations will frame our analysis of the teacher training resources produced by the DG-group. We will look for elements in these resources showing whether the group is aware or not of the necessity of the double instrumental genesis in teachers wishing to integrate DG. We will also try to highlight the way the group orchestrates DG activities both in math classroom and in teacher training.

Potentialities of dynamic geometry

A dynamic geometry environment is computer-based software that allows the user to create geometrical figures and manipulate them into different shapes and positions by dragging their elements, mostly points. One of the distinctive features of DG is that when dragging, the geometrical properties of the figure defined in its construction are preserved. Three main modalities of dragging have been identified (3) dragging for validating/invalidating consists in dragging to check whether the constructed figure preserves its geometrical properties when dragging.

In the analysis of the DG-group resources we will focus on the modalities of dragging in the activities it proposes and especially whether there are changes in the teachers' perceptions of the role of dragging.

DATA ANALYSIS

The analysis of the collected data shows a significant shift in the practices of the DGgroup teachers in three main aspects. The first two are related mostly to their practices as mathematics teachers, the third one to their practices as teacher educators.

Modalities of dragging

Relying on the teachers' auto-analysis of their own practices before and after their involvement into the I2G project [START_REF] Bourgeat | De l'utilisation à la formation : géométrie dynamique et logiciels mathématiques[END_REF], it appears that with their students, the teachers used DG mostly to obtain robust constructions aiming at highlighting invariants in geometric figures:

"Yesterday, obtaining robust geometric constructions and highlighting invariants were the main goals assigned to students: they should construct a figure by using known properties […] which they could validate by the invariance of the figure when dragging."

Nowadays, they propose new types of activities in which they ask students "to explore figures in order to highlight invariants and/or conjecture new properties" (ibid.). The teachers seem to have acknowledged the importance of dragging for conjecturing in students learning and they thus propose various and richer tasks using different modalities of dragging.

The following comment [4], written by one of the DG-group members about a resource in the repository illustrating, with a robust construction, the equality of three ratios in a triangle with a parallel line to one of its sides, shows her awareness of the interest of soft constructions in geometry learning:

"Several improvements are possible: 1. [The point] N can be set free, which will allow visualizing the difference between the case proportional-parallel and the cases where the ratios are not equal".

Although it is difficult to establish a direct link between the questionnaire and the evolution of the teachers' awareness of the DG contributions to the teaching and learning geometry, we can suppose that the numerous discussions about this issue, that eventually led to the definition of criteria related to the added value brought by DG to the math activity, are at the origin of this evolution.

Instrumental orchestration

Regarding the classroom management, the teachers confess to have struggled to combine phases of students' work on computers in a computer lab with collective phases of debate, which often needed to be postponed until the next session in an ordinary classroom, as they say (ibid.):

"Before, the activities with ICT took place in a computer lab in the conditions that postponed the debate and the students-teacher interactions regarding their observations and manipulations in a digital resource."

Nowadays, the teachers orchestrate their ICT-based lessons in a more effective way: the use of a video projector allows articulating individual and collective phases. Indeed, the teachers say: "[Now] we observe the interactions in a genuine triangle "students -teacher -digital resource" (ibid.).

This shift can certainly be related to the pedagogical implementation of the resource, one of the nine aspects that we consider critical for determining the resource quality.

Awareness of the double instrumental genesis

The analysis of the teacher training resources produced by the DG-group reveals that the training programmes the group proposed before 2007, i.e. before its involvement into the I2G project, aimed mostly at helping trainees to master DG system tools. The training activities consisted in series of exercises to solve with DG chosen to illustrate the use of a particular DG tool. Figure 3 The focus of the teacher training programmes in this period was clearly on technical aspects of mastering a DG environment. In terms of instrumental genesis, the DGgroup accompanied trainees' personal geneses of mathematical instruments.

Since 2009, the DG-group teacher training proposals show a significant shift towards considering didactical and pedagogical aspects of DG integration. Indeed, in a training programme proposed in 2009, the group announces the following objective: "The aim of this training programme is to accompany the teacher wishing to take her/his students to a computer lab". After a short phase of solving exercises aiming at getting acquainted with the main DG tools, the trainees are invited to reflect on activities suitable for the DG use, the goal being to bring forward the following aims: introduce a new mathematical concept, construct figures, and put students into a research activity with DG. After having solved a given exercise with DG, the trainees are asked to explore it in light of a possible implementation in a classroom: envisage possible adaptations, anticipate classroom management. Figure 4 shows a training resource, in which the two phases, solving an exercise and exploring it from didactical and pedagogical points of view, are present. The trainees are also led to create their own activities related to a math domain of their choice and adapted to the level of their class. They have to specify the teaching goals and envisage the classroom implementation of the activity.

The DG-group has developed specific resources to help the trainees with this task, such as a description sheet of a session using DG (Fig. 5) or a checklist with questions to ask before using ICT in a classroom, e.g., when to use ICT, do the ICT contributions favour students' learning, or how to integrate an ICT session into an ordinary teaching sequence. These elements show that the DG-group has gotten awareness of the necessity to assist the trainees' professional instrumental geneses so that they can transform DG software not only into a mathematical instrument, but also into a didactical one. Moreover, the resources produced by the group entail signs of the influence of the quality questionnaire, namely considerations of several dimensions such as contributions of DG, didactical exploitation of DG potentialities or instrumental orchestration. This seems to confirm a highly positive impact of the DG-group involvement into a design of DG resource quality questionnaire.

CONCLUSION

In this contribution, we reported about a research on the issue of DG resource quality, conducted by a mixed group of math education researchers and in-service teachers. We attempted to show a positive impact of this collaboration on the teachers' practices both as math teachers and teacher educators. Regarding the use of DG in their classes, we observe significant changes in the nature of tasks the teachers propose to their students: these are richer and more challenging, asking the students to explore figures and conjecture properties, rather than just verify supposed or known properties or validate robust constructions. The teachers are also able to envisage more productive instrumental orchestrations allowing a genuine integration of DG in their math classes. As teacher educators, the group seems to be now much more sensitive to didactical and pedagogical questions related to the DG integration than before. Initially, it focused mostly on technical aspects of mastering DG software, thus accompanying trainees' instrumental geneses yielding a mathematical instrument, whereas nowadays, its programmes include activities allowing the trainees to develop a didactical instrument as well.

NOTES

Figure 1 :

 1 Figure 1: Online questionnaire for reviewing a DG resource in the i2geo repository

Figure 2 :

 2 Figure 2: Schema of the resource quality design methodology

  in the literature (Baccaglini-Frank & Mariotti 2010, Healy 2000, Laborde 2001, Arzarello, Olivero, Paola & Robutti 2002): (1) dragging for verifying consists in dragging to check the presence of the supposed (known) geometrical properties in the figure. According to Hölzl (2001), uses of DG are often limited to this modality, in the sense that students are expected to drag figures to confirm empirically the properties which are more or less given; (2) dragging for conjecturing consists in dragging to look for new properties of the figure through the perception of what remains invariant when dragging;

  shows a typical training activity: the trainees would solve the exercise and indicate what DG tools they have used. Exercise n°4: a) Given a segment [AB], construct a square with [AB] as a side. b) Given a segment [AB], construct a square with [AB] as a diagonal. c) Construct a square with a side [AB] without using « parallel line » and « perpendicular line » tools. N.B. For each question, verify that the construction remains stable.

Figure 3 :

 3 Figure 3: Example of a teacher training activity proposed by DG-group in 2005

  between graphical representation and expression of a first and second degree functionThe trainees open the file « stage1-exo-fct.ggb » They are given the document « stage1-exo-fct.odt » Let the trainees solve the exercise. Ask them to reflect on possible adaptations for a Grade 11 classroom for the next training day. 2 nd day: exploitation of the exercise 3.1 solved during the first day: Ask for possible adaptations. A specific attention should be paid to the method used by the students (successive trials without the properties). The need to review this point with them… how? How to manage this activity?

Figure 4 :

 4 Figure 4: Excerpt of a training plan elaborated by the DG-group in 2011

  Figure 5: DG session description sheet

  1. Interoperable Interactive Geometry for Europe, 2007-2010, http://i2geo.net/xwiki/bin/view/Main/About. 2. i2geo.net 3. Institut de Recherche sur l'Enseignement des Mathématiques (IREM). The IREMs gather together primary, secondary and university teachers to conduct research on problems in math teaching and learning at all school levels, to offer teacher training programmes based on research results, and produce and disseminate pedagogical resources.

4. http://i2geo.net/xwiki/bin/view/Coll_cdording/Egalitedes3rapportsavecunedroiteparalleleauncoteduntriangle?bc=& viewer=comments


INTRODUCTION TO WG15 TECHNOLOGIES AND RESOURCES IN MATHEMATICS EDUCATION

Jana Trgalová 1 , Mirko Maracci 2 , Giorgos Psycharis 3 , Hans-Georg Weigand 4 1 University of Lyon 1, France, 2 University of Pavia, Italy, 3 University of Athens, Greece, 4 Wuerzburg University, Germany Technologies in mathematics education have been a topic of a working group since the CERME 1999. The last two conferences confirmed the relevance of considering information and communication technology (ICT) in mathematics education within a range of various resources, such as software, hand-held devices, online classroom activities, but also more traditional geometry tools or textbooks. A number of important issues related to the design of technologies and resources and their use by students, teachers and teacher educators emerged from discussions within the working group at the previous conferences, such as the importance of coping with a strong interconnectedness of mathematical and technological knowledge in studenttool interactions, or the necessity of including the users' feedback in the ICT and resource design processes [START_REF] Trgalová | CERME7, Introduction to Working Group 15: Technologies and resources in mathematics education[END_REF]. Concerning teacher education questions appeared about the complexity of ICT and resource integration in teachers' practices due to a double instrumental genesis, both personal and professional one [START_REF] Haspekian | The co-construction of a mathematical and a didactical instrument[END_REF]. The idea of teachers' communities, sharing resources and practices in using ICT, emerged as a powerful means to favour teachers' professional development [START_REF] Wenger | Communities of Practice: Learning, Meaning and Identity[END_REF][START_REF] Jaworski | Learning communities in mathematics: creating an inquiry community between teachers and didacticians[END_REF]. These issues require further theoretical and methodological developments.

The call for contributions proposed to deepen these issues in the following three themes: (1) Design and use of technologies and resources, (2) students' learning with technologies and resources, and (3) technologies, resources and teachers' professional development. The group work combined plenary sessions where common issues were discussed, such as theoretical frameworks used in the field of ICT in math education or conceptualization of mathematics with ICT, and parallel sessions addressing various topics, such as software design, task design, teachers' professional development toward the ICT integration, or reports of empirical studies of ICT use. In what follows, we give a brief overview of the working group and the discussions within it, and we outline a few perspectives for the next conference.

WORKING GROUP IN A FEW NUMBERS

The group involved 44 participants from 15 countries: Brazil, Czech Republic, Denmark, France, Germany, Greece, Iceland, Ireland, Israel, Italy, Mexico, Norway, Sweden, Turkey, and United Kingdom. The 27 accepted papers and 6 posters were distributed according to the three themes as follows:

(1) Design and use of technologies and resources 4 papers (2) Students' learning with technologies and resources 13 papers 3 posters (3) Technologies, resources and teachers' professional development 10 papers 3 posters

CONCLUDING REMARKS AND PERSPECTIVES

The papers and posters presented and discussed within the group show a big variety of research topics, theoretical and methodological approaches, which is undoubtedly a sign of a richness of this scientific domain. However, the feeling is that the research presented in these contributions is rather local, focusing on a particular aspect of teaching and/or learning mathematics, rather short-term, not allowing to draw general conclusions about the benefits of ICT in mathematics education, and quite often conducted in controlled, laboratory conditions.

Thus, the participants expressed a need to know more about the "real" use of ICT in mathematics classrooms and outside, but also why ICT is not used. Long-term studies focusing on "ordinary" teachers and "ordinary" classrooms are necessary to explore the impact of the ICT use on students' performances and on teachers' practices. Such studies require developing specific methodologies enabling to assess the effectiveness of ICT in learning processes.

Moreover, research working toward a definition of recommendations or guidelines for teachers suggesting how to use efficiently ICT is missing. For this, surveys of international comparative studies highlighting best practices with ICT use could/should be conducted.

Little or no studies focus on how mathematics as a school subject matter is impacted by the ICT use. We know since many years that computers can perform complex calculations and that educational software provides multiple representations offering reliable visualisations of mathematical concepts. But some "old" questions are still present and valid: Do we still need to teach and learn the same mathematics as before the rise of technology? How can we define the educational added value of ICT use for the learning of mathematics given the explosion of new tools and functionalities provided in technology-rich environments? How can we connect the mathematics embedded within particular ICT tools and curricular mathematics?

In addition, there are emerging research themes in education which are still underrepresented in CERME: the design and use of innovative technologies such as Web 2.0 or mobile technologies, or the design and use of technologies and resources for learners with special educational needs.

Clearly, these concerns could define a research agenda for the coming years. On a CERME working group scale, especially the following two tasks can be carried out:

-Working towards common understanding of theoretical constructs and research vocabulary used in various national contexts. This can be done for example by organizing during the sessions a group work on the analysis of concrete material (e.g. a task, an ICT tool, a set of data) from different theoretical perspectives;

-Working towards capitalising research outcomes, especially from previous technology working groups at CERME conferences. Each contribution to the working group should draw, when relevant, on the results of studies presented at previous conferences.

Table 1 : Distribution of papers and posters in the three themes. REPORT OF THE WORKING GROUP DISCUSSIONS In

 1 spite of several attempts toward building an integrative theoretical framework allowing to address the issues of ICT in math education (e.g. EU project ReMath, Artigue & Mariotti, to appear), the discussions within the WG15 highlighted a variety of theoretical approaches used, such as constructionism (Kynigos & Moustaki), theory of didactic situations (Joubert), instrumental approach (Misfeld), variation theory (Attorps et al.). Abboud-Blanchard & Vandebrouck proposed a new theoretical frame for studying the teachers' professional development, which integrates some of these theories, and Tabach developed a general framework to describe instrumental orchestration. Necessity of networking of theories appeared as an issue. Suggestions on how to proceed were formulated: look at the same set of data with the lens of different frameworks or analyse a given instrumented task from different theoretical viewpoints.

	They described and analysed various means of teacher development, formal (e.g.,
	Akkoç; Santos-Trigo et al.; Sollervall) or informal (Trgalová & Jahn), in presence
	(Balgalmis et al., Dullius) or on-line (Fredriksen; Gravina et al.). The efficiency of
	these different means need to be evaluated, and for this, specific methodologies have
	to be developed. The participants raised a need for a model allowing to analyze the
	evolution of teachers' practices related to the ICT use. Clark-Wilson identified
	perturbations when teachers use technology due to discontinuities within teachers'
	knowledge which she calls "hiccups".

Particular importance was given to software and task design. Lagrange & Psycharis explored the potential of a learning environment and Mackrell et al. designed tasks for a specific environment, both studies drawing on two complementary theoretical frames. Libbrecht & Kortenkamp highlighted the importance of metadata in the design of learning activities and Libbrecht & Zimmermann brought forward didactical design patterns that can impact software construction processes. Pilet et al. designed a piece of software allowing teachers to automatically generate exercises for differentiated instruction adapted to learning needs of various students' groups. Task design was explored by Robová & Vondrová in relation with the integration of netbooks in math classes. Müller proposed an instrument, drawn from history of mathematics, allowing to document changes in the type of tasks proposed in technology environment.

A number of contributions addressed the issue of teacher professional development.

of mathematical concepts with

  specific software was discussed in numerous papers. Students' conceptualisation processes with dynamic geometry were explored by Henning & Hoffkamp (the concept of limit), Kilic (elementary geometry),Kaya et al. (transformational geometry) and Pettersson (linear functions). The use of CAS and handheld calculators was investigated by Persson (development of problem solving skills with TI-Nspire), Rieß & Greefrath (lower achieving students' learning of functions with CAS-calculator), Storfossen (primary school mathematics with graphic calculator) and Grønbaek (use of professional CAS to teach upper secondary mathematics). Weigand explored the interplay between mental, digital and paper representations in a CAS environment in tests and examinations.A general issue of a capitalisation of research results was raised in the light of an important number of research studies on ICT in mathematic education. First steps toward this was done by Bray who provided an overview of recent technological interventions in mathematics education and examined the educational affordances of the technology, and Scheffer who proposed a survey of research conducted by undergraduate, master and doctoral students, aimed at promoting reflections about the ICT integration.

LIST OF PAPERS

Abboud


Sebastien Destercke
email: sebastien.destercke@hds.utc.fr

Multilabel prediction with probability sets: the Hamming loss case

Keywords: Credal sets, multilabel, indeterminate classification, k-nn, binary relevance

In this paper, we study how multilabel predictions can be obtained when our uncertainty is described by a convex set of probabilities. Such predictions, typically consisting of a set of potentially optimal decisions, are hard to make in large decision spaces such as the one considered in multilabel problems. However, we show that when considering the Hamming loss, an approximate prediction can be efficiently computed from label-wise information, as in the precise case. We also perform some first experiments showing the interest of performing partial predictions in the multilabel case.

Introduction

The problem of multi-label classification, which generalizes the traditional (single label) classification setting by allowing multiple labels to belong simultaneously to an instance, has recently attracted a lot of attention. Such a setting indeed appears in a lot of cases: a film can belong to multiple categories, a music can stir multiple emotions [START_REF] Trohidis | Multi-label classification of music into emotions[END_REF], proteins can possess multiple functions [START_REF] Yu | Protein function prediction using dependence maximization[END_REF], etc. In such problems, obtaining a complete ground truth (sets of relevant labels) for the training data and making accurate predictions is more complex than in traditional classification, in which the aim is to predict a single label.

In such a setting the appearance of incomplete observations, i.e., instances for which we do not know whether some labels are relevant, is much more likely. For example, a user may be able to tag a movie as a comedy and not as a science-fiction movie, but may hesitate about whether it should be tagged as a drama. Other examples include cases with high number of labels and where an expert cannot be expected to provide all relevant ones. Such partial labels are commonly called weak labels [START_REF] Sun | Multi-label learning with weak label[END_REF] and are common in problems such as image annotation [START_REF] Tian | Image annotation with weak labels[END_REF] or protein function prediction [START_REF] Yu | Protein function prediction using dependence maximization[END_REF].

Even when considering weak labels, all multilabel methods we are aware of still produce complete predictions as outputs. However, given the complexity of the predictions to make and the likely presence of missing data, it may be sensible to look for means to do cautious yet more trustful predictions. That is it may be interesting for the learner to abstain to make a prediction about a label whose relevance is too uncertain, so that the final prediction is partial but more accurate. Such an approach can be seen as an extension of the reject option implemented in learning problems [START_REF] Bartlett | Classification with a reject option using a hinge loss[END_REF] or of the fact of making partial predictions [START_REF] Corani | Bayesian networks with imprecise probabilities: Theory and application to classification[END_REF], and has been recently investigated for the related problem of label ranking [START_REF] Cheng | Predicting partial orders: ranking with abstention[END_REF][START_REF] Cheng | Label ranking with partial abstention based on thresholded probabilistic models[END_REF].

In this paper, we consider the problem of making partial predictions in the multilabel setting using convex sets of probabilities, or credal sets [START_REF] Levi | The Enterprise of Knowledge[END_REF], as our predictive model. Indeed, making partial predictions is one central feature of approaches using credal sets [START_REF] Corani | Bayesian networks with imprecise probabilities: Theory and application to classification[END_REF], and these approaches are also well-designed to cope with the problem of missing or incomplete data [START_REF] Zaffalon | Exact credal treatment of missing data[END_REF]. However, making partial predictions with credal sets in large decision space such as the one considered in mutlilabel is usually difficult.

In Section 3, we nevertheless demonstrate that when focusing on the Hamming loss, obtaining approximate partial predictions can be done in a quite efficient way by focusing on label-wise information. We then perform (Section 4) some experiments to demonstrate the interest of making partial predictions in the multilabel setting. Section 2 presents necessary background material.

Preliminary material

In this section, we introduce the multilabel setting as well as basic notions needed to deal with sets of probabilities.

Multilabel problem setting

The usual goal of classification problems is to associate an instance x coming from an instance space X to a single (preferred) label of the space Λ = {λ 1 , . . . , λ m } of possible classes. In a multilabel setting, an instance x is associated to a subset L x ⊂ Λ of labels, often called the subset of relevant labels while its complement Λ \ L x is considered as irrelevant. We denote by Y = {0, 1} m the set of m-dimensional binary vectors, and identify a set L of relevant labels with a binary vector y = (y 1 , . . . , y m ) ∈ Y such that y i = 1 if and only if λ i ∈ L. As there is a one-to-one mapping between subsets L of Λ and Y , we will indifferently work with one or the other.

The task in a multilabel problem is the same as in usual classification: to use the training instances (x j , y j ), j = 1, . . . , n to estimate the theoretical conditional probability measure P x : 2 Y → [0, 1] associated to an instance x ∈ X . Ideally, observed outputs y j should be completely specified vectors , however it may be the case that the value for some component y j i is unknown, which will be denoted by y j i = * . We will denote incomplete vectors by capital Y . Alternatively, an incomplete vector Y can be characterized by two sets L ⊆ L ⊆ Λ of necessarily and possible relevant labels, defined as L := {λ i |y i = 1} and L := {λ i |y i = 1 ∨ y i = * } respectively. An incomplete vector Y describes a corresponding set of complete vectors, obtained by replacing each y i = * either by 1 or 0, or equivalently by considering any subset L such that L ⊆ L ⊆ L. To simplify notations, in the sequel we will use the same notation for an incomplete vector and its associated set of complete vectors.

Example 1. Table 1 provides an example of a multilabel data set with In multilabel problems the size of the prediction space increases exponentially with m (|Y | = 32768 for m = 15), meaning that estimating directly P x will be intractable even for limited sizes of Λ . As a means to solve this issue, different authors have proposed so-called transformation techniques [START_REF] Tsoumakas | Multi-label classification: An overview[END_REF] that reduce the initial problem (in which 2 m parameters have to be estimated) into a set of simpler problems. For instance Binary Relevance (BR) consists in predicting relevance label-wise, solving an independent binary problem for each label. It therefore comes down to estimate m parameters P x (y i ), i = 1, . . . , m and to predict ŷi = 1 if P x (y i = 1) ≥ 1 /2. A common critic of the BR approach is that it does not take account of label dependencies, however it has been shown that this approach is theoretically optimal for the Hamming loss, on which this paper focuses [START_REF] Dembczynski | On label dependence and loss minimization in multi-label classification[END_REF]. Other reduction approaches include, for instance, Calibrated Ranking (CR) [START_REF] Fürnkranz | Multilabel classification via calibrated label ranking[END_REF] that focuses on pairwise comparisons.

Λ = {λ 1 , λ 2 , λ 3 }. Y 3 = [ * 1 0] is an incomplete observed instance with L 3 = {λ 2 } and L 3 = {λ 1 , λ 2 }. Its corresponding set of complete vectors is {[0 1 0], [1 1 0]} X 1 X 2 X 3 X
Another issue is that making a precise and accurate estimation of P x is an extremely difficult problem given the number 2 m of alternatives and the possible presence of missing data. This problem is even more severe if little data are available, and this is why making cautious inferences (i.e., partial predictions) using as model a (convex) set P x of probability distributions may be interesting in the multilabel setting.

Notions about probability sets

We assume that our uncertainty is described by a convex set of probabilities P x defined over Y rather than by a precise probability measure P x . Such a set is usually defined either by a collection of linear constraints on the probability masses or by a set of extreme probabilities. Given such a set, we can define for any event A ⊆ Y the notions of lower and upper probabilities P x (A) and P x (A), respectively defined as

P x (A) = inf P x ∈P x P x (A) and P x (A) = sup P x ∈P x P x (A).
Lower and upper probabilities are dual, in the sense that P(A) = 1 -P(A c ). Similarly, if we consider a real-valued bounded function f : Y → R, the lower and upper expectations E x ( f ) and E x ( f ) are defined as

E x ( f ) = inf P x ∈P x E x ( f ) and E x ( f ) = sup P x ∈P x E x ( f ),
where E x ( f ) is the expectation of f w.r.t. P. Lower and upper expectations are also dual, in the sense that E( f ) = -E(f ). They are also scale and translation invariant in the sense that given two numbers

α ∈ R + , β ∈ R, we have E(α f + β ) = αE( f ) + β
In the next sections, we explore how the multilabel problem can be solved with such credal estimates. We discuss the problem, usually computationally intensive, of making partial decision and show that it can be simplified when considering the Hamming loss as our loss function. Using these results, we then perform some experiment based on label-wise decomposition and k-nn algorithms to assess the interest of making partial predictions based on credal sets.

Credal multilabel predictions with Hamming loss

In this section, we first recall the principle of credal predictions, before proceeding to show that in the case of Hamming loss, such predictions can be efficiently approximated by an outer-approximation.

Credal predictions

Once a space Y of possible observations is defined, selecting a prediction, or equivalently making a decision, requires to define:

a space A = {a 1 , . . . , a d } of possible actions (sometimes equal to Y , but not necessarily); a loss function ℓ : A × Y → R such that ℓ(a, y) is the loss associated to action a when y is the ground-truth.

Given an instance x and a precise estimate Px , a decision a will be preferred to a decision a ′ under loss function ℓ, denote a ≻ ℓ a ′ , if

E x ℓ(a ′ , •) -ℓ(a, •) = ∑ y∈Y Px (y) ℓ(a ′ , y) -ℓ(a, y) > 0, (1)
where E x is the expectation w.r.t. Px . This equation means that exchanging a ′ for a would incur a positive expected loss, therefore a should be preferred to a ′ . In the case of a precise estimate Px , ≻ ℓ is a complete pre-order and the prediction comes down to take the maximal element of this pre-order, i.e., âℓ = arg min

a∈A E x (ℓ(a, •)) = arg min a∈A ∑ y∈Y Px (y)ℓ(a, y) (2)
that is to minimize the expected loss (ties can be broken arbitrarily, as they will lead to the same expected loss). This means that finding the best action (or prediction) will therefore requires d computations of expectations.

When considering a set P x as cautious estimate, there are many ways [START_REF] Troffaes | Decision making under uncertainty using imprecise probabilities[END_REF] to extend Eq. ( 1), but the most well-founded is the maximality criterion, which states that a

≻ ℓ a ′ , if E x ℓ(a ′ , •) -ℓ(a, •) > 0, (3)
that is if exchanging a ′ for a is guaranteed to give a positive expected loss. In such a case, the relation ≻ ℓ will be a partial order, and the maximal set Âℓ of alternatives will be chosen as a prediction, that is

Âℓ = {a ∈ A | ∃a ′ ∈ A s.t. a ′ ≻ ℓ a}. ( 4
)
Computing Âℓ requires at worst d(d -1) computations, a quadratic number of comparisons with respect to the number of alternatives. Also notes that evaluating Eq. ( 3) usually requires solving a linear programming problem, a computationally more intensive task than evaluating Eq. ( 1). Âℓ is a cautious prediction, since it considers a set of potential optimal solutions. Multilabel loss functions usually considers the set A = Y as possible actions, or even bigger sets (for example the ranking loss considers as actions the set of complete orders over Λ ). This means that getting âℓ is already quite hard in the general case, hence computing Âℓ will be intractable in most cases, as the worst number of computation will then be 2 2m (m = 15 labels means at worst ∼ 10 9 comparisons).

In the next subsection, we show that for the Hamming loss ℓ H , we can get an outer approximation of Âℓ at an affordable computational cost. Offering such efficient way to make cautious predictions based on P x is essential to be able to use such kind of models in complex problems.

The Hamming loss

Let the set of alternatives be A = Y . Given an observation y and a prediction ŷ, Hamming loss ℓ H reads

ℓ H ( ŷ, y) = 1 m m ∑ i=1 1 ( ŷi =y i ) . (5)
It counts the number of labels for which our prediction is wrong, and normalizes it.

When the estimate Px is precise, it is known [START_REF] Dembczynski | On label dependence and loss minimization in multi-label classification[END_REF] that the optimal decision is the vector ŷ such that ŷ j = 1 if Px (y j = 1) ≥ 1 /2 and ŷ j = 0 else. In particular, this means that the optimal decision can be derived from the sole knowledge of the marginals Px (y j = 1), j = 1, . . . , n, provided they are good estimates of P x . Given a probability set P x , let Ŷℓ H be the maximal set of vectors that would be obtained using Eq. ( 4). The next proposition shows that Ŷℓ H can be outer-approximated using the marginals of the cautious estimate P x , in contrast with the precise case.

Proposition 1. Let P x be our estimate, then the imprecise vector Ŷ * such that

Ŷ * j =      1 if P(y j = 1) > 1 /2 0 if P(y j = 0) > 1 /2
* else, i.e. P(y j = 1) ≤ 1 /2 ≤ P(y j = 1)

for j = 1, . . . , m
is an outer approximation of Ŷℓ H , in the sense that Ŷℓ H ⊆ Ŷ * .

Proof. Consider a given j ∈ {1, . . . , m} and two alternatives ŷ and ŷ′ such that ŷ j = 1 = ŷ′ j and ŷi = ŷ′ i for any i = j. Then for any y such that y j = 1 we have

ℓ H ( ŷ′ , y) -ℓ H ( ŷ, y) = ∑ k = j 1 ( ŷ′ k =y k ) + 1 ( ŷ′ j =y j ) -∑ k = j 1 ( ŷk =y k ) + 1 ( ŷ j =y j ) = 1 ( ŷ′ j =0) -1 ( ŷ j =0) = 1,
and for any y such that y j = 0 we have

ℓ H ( ŷ′ , y) -ℓ H ( ŷ, y) = ∑ k = j 1 ( ŷ′ k =y k ) + 1 ( ŷ′ j =y j ) -∑ k = j 1 ( ŷk =y k ) + 1 ( ŷ j =y j ) = 1 ( ŷ′ j =1) -1 ( ŷ j =0) = -1. We therefore have (ℓ H ( ŷ′ ,•)-ℓ H ( ŷ,•)+1) /2 = 1 (y j =1) , hence P(y j = 1) = E ℓ H ( ŷ′ , •) -ℓ H ( ŷ, •) + 1 2 = 1 2 E ℓ H ( ŷ′ , •) -ℓ H ( ŷ, •) + 1 2
the last equality coming from scale and translation invariance. Hence E(ℓ H ( ŷ′ , •)ℓ H ( ŷ, •)) > 0 if and only if P(y j = 1) > 1 /2. This means that, if P(y j = 1) > 1 /2, any vector ŷ′ with ŷ′ j = 0 is dominated (in the sense of Eq. ( 3)) by the vector ŷ where only the j-th element is modified, hence no vector with ŷ′ j = 0 is in the maximal set Ŷℓ H . The proof showing that if P(y j = 0) > 1 /2, then no vector with ŷ′ j = 1 is in the maximal set is similar. We now provide an example showing that the inclusion can be strict in general.

Example 2. Consider the 2 label case Λ = {λ 1 , λ 2 } with the following constraints:

0.4 ≤ P(y 1 = 1) = P({[1 0]}) + P({[1 1]}) ≤ 0.6 0.9 (P({[1 0]}) + P({[1 1]})) = P({[1 0]}) 0.84 (P({[0 1]}) + P({[0 0]})) = P({[0 1]})
These constraints describe a convex set P, whose extreme points (obtained by saturating the first inequality one way or another) are summarized in Table 2. The first constraints induces that P(y 1 = 1) = 0.4 and P(y 1 = 0) = 0.6, while the bounds P(y 2 = 1) = 0.396, P(y 2 = 1) = 0.544, are reached by the extreme distributions P([1 1]) = 0.06, P([0 1]) = 0.336 and P([1 1]) = 0.04, P([0 1]) = 0.504, respectively. Given these bounds, we have that Ŷ * = [ * * ] corresponds to the complete space Y (i.e., the empty prediction). Yet we have that

E (ℓ H ([1 1], •) -ℓ H ([0 0], •)) = 0.0008 ≥ 0
also obtained with the distribution P([1 1]) = 0.06, P([0 0]) = 0.064. This means that the vector [0 0] is not in the maximal set Ŷℓ H , while it is included in Ŷ * .

Proposition 1 shows that we can rely on marginal information to provide an outerapproximation of Ŷℓ H that is efficient to compute, as it requires to compute 2m values, which are to be compared to the 2 2m usually required to assess Ŷℓ H . It also indicates that extensions of the binary relevance approach are well adapted to provide partial predictions from credal sets when considering the Hamming loss, and that in this case global models integrating label dependencies are not necessary, thus saving a lot of heavy computations.

P({[0 0]}) P({[1 0]}) P({[0 1]}) P({[1 1]}) 0.

First experimentations

In this section, we provide first experimentations illustrating the effect of making partial predictions with a decreasing amount of information. These experiments illustrate that such partial predictions may indeed improve the correctness of predictions.

Evaluation

Usual loss functions such as Eq. ( 5) are based on complete predictions. When making partial predictions, such loss functions need to be adapted. This can be done, for instance, by decomposing it into two components [START_REF] Cheng | Predicting partial orders: ranking with abstention[END_REF], one measuring the accuracy or correctness of the made prediction, the other measuring its completeness.

If the partial prediction is an incomplete vector such as Ŷ * , then Hamming loss can be easily split into these two components. Given the prediction Ŷ * characterized by subsets L, L, let us denote Q = Λ \ (L ∩ L) the set of predicted labels (i.e., labels such that Ŷ * j = 1 or Ŷ * j = 0). Then, if the observed set is y, we define correctness (CR) and completeness (CP) as

CR( Ŷ * , y) = 1 |Q| ∑ λ i ∈Q 1 ( ŷi =y i ) ; (6)
CP( Ŷ * , y) = |Q| m . (7)
when predicting complete vectors, then CP = 1 and CR equals the Hamming loss [START_REF] Dembczynski | On label dependence and loss minimization in multi-label classification[END_REF]. When predicting the empty vector, then CP = 0 and by convention CR = 1.

Method

The method we used was to apply, label-wise, the k-nn method using lower probabilities introduced in [START_REF] Destercke | A k-nearest neighbours method based on imprecise probabilities[END_REF] (in which details can be found). This means that from an initial training data set D, m data sets D j corresponding to binary classification problems are built, this decomposition being illustrated in Figure 1. Given an instance x, the result of the k-nn method on data set D j provides an estimate of [P(y j = 1), P(y j = 1)] and by duality an estimate of P(y j = 0) = 1 -P(y j = 1) and P(y j = 0) = 1 -P(y j = 1). The method also automatically takes account of missing label information, and treat such missing data in a conservative way, considering them as completely vacuous information (that is, we treat them as non-MAR variables [START_REF] Zaffalon | Conservative inference rule for uncertain reasoning under incompleteness[END_REF]).

Results

In the experiments, the parameters of the k-nn algorithm were set to β = 0.75 and ε 0 = 0.99, so that results obtained when fixing the number k of neighbors to 1 display a sufficient completeness. ε 0 settles the initial imprecision, while β determines how much imprecision increases with distance (details about the role of these parameters can be found in [START_REF] Destercke | A k-nearest neighbours method based on imprecise probabilities[END_REF]). We ran experiments on well-known multilabel data sets having real-valued features. Their characteristics are summarized in Table 3. For each of them, we ran a 10-fold cross validation with the number k of neighbors varying from 1 to 5, and with various percentages of missing labels in the training data set (0%, 20% and 40%). Varying k in the algorithm allows us to control the completeness of the prediction: the higher k is, the more imprecise become the estimations. The results of the experiment are displayed in Figure 2. From this figure, two main conclusions can be drawn: on the used data sets, allowing for partial predictions (here, by increasing the number k of neighbours) systematically improve the correctness, and missing labels only influence the completeness of the predictions, not the correctness of the results. This latter fact, however, may be due to the learning method. How fast completeness decreases with the number of neighbors, however, clearly depends on the data set.

Conclusions

Producing sets of optimal predictions in the multilabel setting when uncertainty is modeled by convex probability sets is computationally hard. The main contribution of this paper was to show that when using the Hamming loss, such sets can be easily outerapproximated by focusing only on the marginal probability bounds of each label being relevant. This makes both computation and learning issues easier, as one can focus on estimating such marginals (instead of the whole joint model). We can consider that as an important result, as it shows that imprecise probabilistic approaches are computationally affordable (at least under some conditions). We also made some first preliminary experiments indicating the interest of producing such partial predictions, showing that making more cautious predictions lead to more correct predictions. In the future, we intend to make similar studies for other well-known loss functions, such as the ranking loss. We also intend to make further the experiments, i.e., to compare this approach with other methods, or to empirically assess (for small values of m) the quality of the made approximation.

Table 1 .

 1 Multilabel data set example

	4 y 1 y 2 y 3
	107.1 25 Blue 60 1 0 0
	-50 10 Red 40 1 0 1
	200.6 30 Blue 58 * 1 0
	107.1 5 Green 33 0 1 *
	. . . . . . . . . . . . . . . . . . . . .

Table 2 .

 2 Extreme points of P of Example 2

	096	0.36	0.504	0.04
	0.064	0.54	0.336	0.06

  X 1 X 2 X 3 X 4 y 1 y 2 y 3 107.1 25 Blue 60 1 0 0 -50 10 Red 40 1 0 1 200.6 30 Blue 58 * 1 0 . . . . . . . . . . . . . . . . . . . . .

	data set D 1	data set D 2	data set D 3
	X 1 . . . X 4 y 1	X 1 . . . X 4 y 2	X 1 . . . X 4 y 3
	107.1	60 1	107.1	60 0	107.1	60 0
	-50	40 1	-50	40 0	-50	40 1
	200.6	58 *	200.6	58 1	200.6	58 0
	. . .	. . . . . .	. . .	. . . . . .	. . .	. . . . . .
			Fig. 1. Label-wise decomposition of data set D

Table 3 .

 3 Multilabel data sets summary

	Name	# Features	# Labels	# Instances
	emotion	72	6	593
	scene	294	6	2407
	yeast	103	14	2417
	CAL500	68	174	502

Acknowledgements

Work carried out in the framework of the Labex MS2T, funded by the French Government, through the National Agency for Research (Reference ANR-11-IDEX-0004-02)


Matthias C M Troffaes

Frank P A Coolen

Sébastien Destercke

A Note on Learning Dependence Under Severe Uncertainty

Keywords: bivariate data, categorical data, copula, Gaussian copula, robust Bayesian, imprecise probability

We propose two models, one continuous and one categorical, to learn about dependence between two random variables, given only limited joint observations, but assuming that the marginals are precisely known. The continuous model focuses on the Gaussian case, while the categorical model is generic. We illustrate the resulting statistical inferences on a simple example concerning the body mass index. Both methods can be extended easily to three or more random variables.

Introduction

Sklar's theorem [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] states that any multivariate distribution of k variables can be expressed through a density on [0, 1] k with uniform marginals-this density is called a copula-and the marginal distributions of each of the variables. For this reason, copulas [START_REF] Roger | An introduction to copulas[END_REF] have become an indispensible tool to model and learn statistical dependence in multivariate models: they allow of estimation of the dependence structure, separately from the marginal structure.

Estimating dependence requires joint observations, which in many cases are only available in small amounts, while substantial amounts of marginal data may be available. For example, when studying the reliability of a system, it is common to have good information about the reliability of each system component, yet to have only little information about joint failures [START_REF] Matthias | A robust data driven approach to quantifying common-cause failure in power networks[END_REF]. Imprecise probabilities provide one possible theoretical basis for dealing with small sample sizes, by representing knowledge as a set of distributions [START_REF] Boole | An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities[END_REF][START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF][START_REF] Augustin | Introduction to imprecise probabilities[END_REF], rather than a single distribution necessarily based on somewhat arbitrary assumptions [START_REF] Wasserman | The selection of prior distributions by formal rules[END_REF].

Copulas and imprecise probabilities have been studied in the literature by various researchers. The Fréchet-Hoeffding copula bounds, which represent completely unknown dependence, are used for instance in probabilistic arithmetic [START_REF] Williamson | Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds[END_REF] and p-boxes [START_REF] Scott Ferson | Constructing probability boxes and Dempster-Shafer structures[END_REF][START_REF] Matthias | Probability boxes on totally preordered spaces for multivariate modelling[END_REF]. One theoretical difficulty is that there is no straightforward imprecise equivalent of Sklar's theorem, say, expressing any set of joint distributions as a sets of copulas along with a set of marginal distributions [START_REF] Pelessoni | Imprecise copulas and bivariate stochastic orders[END_REF]: it appears that, when working with sets of distributions, separating the dependence structure from the marginal structure is a lot more difficult in general.

In this paper, we propose and investigate a few statistical models for robust dependence learning from limited data. We state an imprecise version of Sklar's theorem when marginal distributions are fully known, separating precise marginals from the imprecise dependence structure. We propose a range of parametric models for the bivariate categorical case. Finally, demonstrate our findings on a toy example: estimating the body mass index from height and mass data.

Section 2 explores a continuous model, focusing on the multivariate normal model, while section 3 provides a first exploration of a generic categorical model.

Robust Bayesian Correlation Learning for Bivariate Normal Sampling

We start with revising a simple and well-studied model: sampling from the bivariate normal distribution. We will derive some new results that are relevant to dependence learning. Our analysis starts from Quaeghebeur and De Cooman's robust Bayesian framework for sampling from the exponential family [START_REF] Quaeghebeur | Imprecise probability models for inference in exponential families[END_REF].

Inference with known Mean and Unknown Covariance Matrix

Let Z i := (Z i1 , . . . , Z ik ) be a multivariate normally distributed random variable with known mean-which we can assume to be zero without loss of generality through translation of the data-but unknown covariance matrix Σ ∈ R k×k . A particular realisation of Z i is denoted by a lower case letter z i := (z i1 , . . . , z ik ) ∈ R k . The likelihood of an i.i.d. sample z 1 , . . . , z n is:

f (z 1 , . . . , z n | Σ) ∝ |Σ| -n/2 n i=1 exp - 1 2 z T i Σ -1 z i (1) = |Σ| -n/2 exp - 1 2 n i=1 tr z i z T i Σ -1 (2)
where the data z i ∈ R k are organised as row vectors, so z i z T i is the matrix containing z iℓ z iℓ ′ in row ℓ and column ℓ ′ .

A family of conjugate priors for this density is the family of inverse Wishart distributions with hyperparameters ν 0 > 0 and Ψ 0 ∈ R k×k positive definite [START_REF] Bernado | Bayesian Theory[END_REF]:

f (Σ | ν 0 , Ψ 0 ) ∝ |Σ| -ν 0 +k+1 2 exp - 1 2 tr Ψ 0 Σ -1 (3)
The posterior distribution is obtained by updating the hyperparameters through:

ν n = ν 0 + n Ψ n = Ψ 0 + n i=1 z i z T i (4)
The prior expected covariance matrix is given by

E (Σ | ν 0 , Ψ 0 ) = Ψ 0 ν 0 -k -1 =: Σ 0 (5)
and therefore, through conjugacy, the posterior expected covariance matrix is

E (Σ | z 1 , . . . , z n , ν 0 , Ψ 0 ) = E (Σ | ν n , Ψ n ) = Ψ n ν n -k -1 (6) = Ψ 0 + n i=1 z i z T i ν 0 + n -k -1 (7) = (ν 0 -k -1)Σ 0 + n i=1 z i z T i ν 0 + n -k -1 =: Σ n (8)
For robust Bayesian analysis aiming to learn about the dependence between two random variables, we now need to identify a reasonable set of prior distributions, or, in our conjugate setting, a reasonable set of hyperparameters ν 0 and Ψ 0 .

The formula for the posterior expected covariance matrix shows that ν 0 determines our learning speed, that is, how many observations n we need before starting to move towards our data. So, ν 0 is similar to the s value in the imprecise Dirichlet model [START_REF] Walley | Inferences from multinomial data: Learning about a bag of marbles[END_REF]. Here too, we will simply assume ν 0 to be fixed to whatever value is judged to lead to a reasonable learning speed. For fixed ν 0 , any particular choice of Ψ 0 corresponds to a prior covariance matrix Σ 0 .

Let us now study the bivariate case (k = 2) in more detail. We will write X i for Z i1 and Y i for Z i2 . We would choose

Ψ 0 = (ν 0 -3) σ 2 X ρ 0 σ X σ Y ρ 0 σ X σ Y σ 2 Y ( 9
)
if we had prior variances σ X > 0 and σ Y > 0 for the two components, as well as the prior correlation coefficient ρ 0 ∈ [-1, 1]. For this paper focusing on dependence, we are mostly interested in cases where the marginals are well known, i.e. well known prior σ X and σ Y , but unknown prior correlation ρ 0 . We will therefore study the set of priors with all parameters fixed, except for ρ 0 , which we assume to be vacuous a priori. Without loss of generality, by rescaling, we can assume that σ X = σ Y = 1, leaving us with just two hyperparameters:

ν 0 > 0 and ρ 0 ∈ [-1, 1].
The posterior covariance matrix becomes:

Σ n = 1 ν ′ 0 + n ν ′ 0 + n i=1 x 2 i ν ′ 0 ρ 0 + n i=1 x i y i ν ′ 0 ρ 0 + n i=1 x i y i ν ′ 0 + n i=1 y 2 i ( 10
)
where

ν ′ 0 = ν 0 -k-1 = ν 0 -3.
Provided that the sample variance is approximately equal to the prior variance, i.e. n i=1

x 2 i ≈ nσ 2 X = n n i=1 y 2 i ≈ nσ 2 Y = n (11)
our expression for Σ n becomes:

1 ρ n ρ n 1 ( 12
)
where

ρ n = ν ′ 0 ρ 0 + n i=1 x i y i ν ′ 0 + n . (13)
Equation ( 12) is the covariance matrix of a bivariate normal with unit marginal variances and correlation coefficient ρ n . For vacuous prior correlation, ρ 0 ∈ [-1, 1], we thus get the following posterior bounds on the correlation:

ρ n = -ν ′ 0 + n i=1 x i y i ν ′ 0 + n (14)
ρ n = ν ′ 0 + n i=1 x i y i ν ′ 0 + n (15)
provided that our observations, and our prior, have unit variance and zero mean (which we can achieve by linear transformation without loss of generality). This analysis generalises almost trivially to cases with more than two variables, k > 2-we leave this to the reader.

Application to body mass index example

We now illustrate our model on the evaluation of the body mass index R = X/Y 2 , where X is a person's weight in kilograms and Y is his/her height in meters. The body mass index is commonly used to detect under-and overweight. We aim [START_REF] Augustin | Introduction to imprecise probabilities[END_REF] to assess the dependence between X and Y in a particular population, and (2) to extract a robust inference about R in this population.

We consider 30 paired observations of heights and weights of 11 years old girls [5, p. 75]. The weight X has sample mean x = 36.2, sample standard deviation s X = 7.7, with no strong evidence against normality (p-value 3 0.017). The height Y has sample mean ȳ = 1.448, sample standard deviation s Y = 0.077, with no evidence against normality whatsoever (p-value 0.711). We will assume that X and Y have known means, equal to x and ȳ. We also assume that, a priori, σ X = 7.7 and σ Y = 0.077 in eq. ( 9), but we are vacuous about the prior correlation. For reference, it may be useful to note that the sample correlation between X and Y is in fact 0.742. For the sake the example, we assume that the sample is drawn from a bivariate normal distribution, although there is reasonably strong evidence against joint normality (p-value 0.00388).

Figure 1 shows the bounds on the correlation of the posterior covariance matrix in eq. [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF]

p (x | θ) = θ x , p (y | φ) = φ y . (16)
We assume that m X ≥ 2, m Y ≥ 2, θ x > 0 and φ y > 0.

We are interested in learning the dependence structure of X and Y . One very general way to express the full joint distribution of (X, Y ) is by introducing parameters w xy such that respectively. This model specification is over-parametrized, but it allows us to model the marginal distributions and the dependence structure separately, where the matrix w plays precisely a similar role as a copula in general bivariate models for (usually) continuous random quantities. However, a key difference and major difficulty with the above model is that the constraints on w depend on θ and φ: the separation is thus not as complete as with copulas, where dependence structure is parametrised independently of the marginals. For this reason, it seems most natural to consider a two-stage situation where we first learn about the marginal parameters θ and φ, followed by learning about the dependence structure w conditional on what we learnt about θ and φ.

Inference for Known Marginals

For this reason, as a stepping stone towards general inference about θ, φ, and w, here we consider a scenario where the marginal distributions are already fully known, and we only aim at inference about w. While this may appear somewhat restrictive, and perhaps even artificial, there are practical scenarios where one has very substantial information about the probability distributions for the random quantities X and Y separately, but relatively little information about their joint distribution.

There are (m X -1)(m Y -1) degrees of freedom for the components of w. In case data is limited, to enable sufficiently useful inference, it seems natural to assume a reduced-dimensional parametric form for w, which may naturally correspond to an (assumed) ordering of the categories, as we will illustrate in section 3.

A problem is that this estimate will usually violate eqs. ( 19) and (20). For instance,

x∈X ŵxy θ x = x∈X n xy nθ x φ y θ x = n * y nφ y = 1 (25)
as soon as n * y n = φ y . A proper maximum likelihood estimate would maximize the likelihood subject to all constraints embodied by eqs. ( 18) to (20). Solving this optimisation problem poses an interesting challenge.

Bayesian inference for w will face a similar challenge: w lives in a convex subspace of R m X ×m Y determined by eqs. ( 18) to (20). Application of Bayes's theorem requires integrating (or simulating) over this space. Nevertheless, the basic principles behind Bayesian inference for w are simple, and sensitivity analysis is similar to the imprecise Dirichlet model [START_REF] Walley | Inferences from multinomial data: Learning about a bag of marbles[END_REF]. Specific dimension-reduced models, where we have a much better handle on the parameter space, will be illustrated in more detail in section 3.3.

The likelihood is given by

x∈X y∈Y (w xy θ x φ y ) nxy (26)
so as conjugate prior we can choose

f (w | α 0 ) ∝ g(w) x∈X y∈Y (w xy θ x φ y ) α0xy (27)
where α 0xy > 0 and g is some arbitrary non-negative function (as long as the right hand side integrates to a finite value). With ν 0 := xy α 0xy , this prior distribution can be interpreted as reflecting prior information equivalent to ν 0 observations of which α 0xy were (X, Y ) = (x, y). The corresponding posterior distribution is clearly

f (w | α n ) with α nxy = α 0xy + n xy .
Sensitivity analysis on this model could then follow an approach similar to Walley's imprecise Dirichlet model [START_REF] Walley | Inferences from multinomial data: Learning about a bag of marbles[END_REF], by taking the set of all prior distributions for a fixed value of ν 0 . In case of an informative set of prior distributions, one may also allow the value of ν 0 to vary within a set to allow prior-data conflict modelling [START_REF] Walter | Imprecision and prior-data conflict in generalized Bayesian inference[END_REF].

As already mentioned, the remaining key difficulty is to integrate the conjugate density over the parameter space. For this reason, in the next section, we consider a reduced model.

Reduced Model

As a first and basic example of a reduced parametric form, consider the case X = Y = {1, 2, 3} with known θ x = φ y = 1/3 for all x and y ∈ {1, 2, 3}. If the categories are ordered in some natural manner, then it might be quite reasonable to specify

w =   1 + 2α 1 -α 1 -α 1 -α 1 + 2α 1 -α 1 -α 1 -α 1 + 2α   (28)
with α ∈ [0, 1]. It is easily verified that this model satisfies eqs. ( 18) to (20): the full matrix sums to 9, and each of the rows and colums sum to 3. Note that there is no logical requirement to avoid α ∈ [-1/2, 0), we just use this small example as an illustration. In this model, α = 0 corresponds to full independence between X and Y , whereas α = 1 corresponds to perfect correlation X = Y . Therefore, this corresponds to a scenario where we may suspect positive correlation between X and Y , but we are unsure about the strength of correlation. Note that the actual model reduction is achieved by additionally assuming that X = Y = x has the same probability for all x ∈ {1, 2, 3}, and similar for X = x ∩ Y = y for all x = y.

With these assumptions, statistical inference is concerned with learning about the parameter α ∈ [0, 1]. The likelihood function is

(1 + 2α) t (1 -α) n-t (
= 3t-n 2n if 3t ≥ n, 0 otherwise. ( 30
)
For a Bayesian approach to inference for this model, we can define a conjugate prior

f (α | ν 0 , τ 0 ) ∝ (1 + 2α) τ0 (1 -α) ν0-τ0 (31)
with τ 0 ∈ [0, ν 0 ], with the possible interpretation as reflecting prior information which is equivalent to ν 0 observations of which τ 0 have X = Y . The posterior distribution is simply f (α | ν 0 + n, τ 0 + t) . Sensitivity analysis is again straightforward by taking the set of prior distributions consisting of all with for example τ 0 ∈ [0, ν 0 ] for a fixed ν 0 . For instance, we would get the following robust estimate for the posterior mode of α:

αn = 3t -ν 0 -n 2(ν 0 + n) , 3t + 2ν 0 -n 2(ν 0 + n) (32)
when 3t ≥ ν 0 + n, with similar formulas when 3t < ν 0 + n (truncating negative values to zero).

Application to body mass index example

To apply the categorical model to our data, we must first discretize them, with the ordering of the categories following the ordering of natural numbers. To obtain three categories with uniform marginals, we simply discretized the 99% prediction intervals of each Gaussian marginals of section 2. Figure 2 shows the bounds on the posterior mode αn in eq. ( 32), with ν 0 = 2. The results are similar to those obtained in section 2.2, showing that even this very simple discretized model can capture the correlation between X and Y , with the bounds on α30 being [0.56, 0.66]. From these values and the bounds of the categories, we can easily obtain bounds on the expectation of R: E(R) = 12.9 and E(R) = 22.4, which gives a much wider interval than in section 2.2. This is due to the very rough discretization: a finer discretization would likely provide a narrower interval. The lower and upper probabilities of A = [14, 19.5]

Conclusion

In this paper, we have introduced two preliminary models-a continuous one and a discrete one-to model dependence when joint data is limited, but assuming that the marginals are precisely known. The continuous model focused on the very special multivariate normal case. However, already in our simple example, we have seen that joint normality is rarely satisfied in practice. A major challenge is to provide methods for dependence modelling, that are both flexible and computationally tractable, whilst still producing useful inferences.

Even though the models and example studied are very preliminary, we feel that extensions of the discrete model could provide more flexibility, whilst still being easy to learn and to compute with. We see it as a promising path to learn dependency structures with imprecise probabilistic models. In particular, it can be seen as a way to approximate a continuous model, as we did in the example. In the future we plan to work on such extensions and on the identification of parametric matrices of weights more flexible than the reduced one presented here.

Finally, an obvious extension to the present work would be to relax the assumption that marginals are precisely identified, and to work with sets of marginals instead. However, this raises challenging theoretical issues, as defining a well-founded extension or equivalent formulation of Sklar's theorem for imprecise models is far from trivial.

Fig. 1 .

 1 Fig. 1. Lower and upper correlation estimates ρ n and ρ n as a function of the sample size n, with ν ′ 0 = 2. The solid horizontal line denotes the sample correlation for n = 30.

psubject to the constraints x∈X y∈Y w xy θ x φ y = 1

 1 (x, y | θ, φ, w) = w xy θ x φ y (17)

y∈Y w xy φ y = 1

 1 and (20) simply follow from x∈X w xy θ x φ y = x∈X p (x, y | θ, φ, w) = p (y | θ, φ, w) = φ y (21) y∈Y w xy θ x φ y = y∈Y p (x, y | θ, φ, w) = p (x | θ, φ, w) = θ x (22)

3 .

 3 Let n xy denote the number of observations of (X, Y ) = (x, y), with total number of observations n = x∈X y∈Y n xy and row and column totals denoted by n x * = y∈Y n xy and n * y = x∈X n xy , respectively. So, there are n x * observations of X = x and n * y observations of Y = y.Without further restrictions on w, it seems tempting to fit the model to match the non-parametric maximum likelihood estimate p (x, y | w)

  are this time P (R ∈ A) = 0 and P (R ∈ A) = 0.96, which are almost vacuous and again show that to have meaningful inferences, a finer discretization is needed.

Fig. 2 .

 2 Fig. 2. Bounds on αn as a function of the sample size n, with ν0 = 2.

  29) with t = n 11 + n 22 + n 33 and n = xy n xy as before. The maximum likelihood estimate is α