INDEPENDENT NATURAL EXTENSION FOR CHOICE FUNCTIONS

A BSTRACT . We introduce an independence notion for choice functions, which we call ‘epistemic independence’ following the work by De Cooman et al. [17] for lower previsions, and study it in a multivariate setting. This work is a continuation of earlier work of one of the authors [29]


Introduction
The framework of sets of desirable gamble sets has incredible expressive power for representing imprecise probabilistic beliefs. A strict generalization of the more familiar framework of sets of desirable gambles, it also allows us to model preferences that are not expressible merely by making binary comparisons between the available options. The dark side of this great power is that sets of desirable gamble sets are very difficult to work with; it will generally be conceptually and computationally intractable to begin modelling some credal state representing many variables with a set of desirable gamble sets that expresses the agent's beliefs about the entire possibility space. However, it will frequently be a much more manageable starting point to begin with some simpler local assessments about various particular ways of partitioning up the possibility space; the utility of this approach is greatly dependent on our ability to usefully combine these assessments into models of the entire space.
The first step is to show how to combine local assessments on uncertain variables that the agent regards as inde-pendent, and this is the project of our paper. We investigate the least informative way of combining these assessments into a joint set of desirable gamble sets, called the independent natural extension. The first few sections of the paper are devoted to developing the formal and notational details that we will need to represent the various concepts at work, recapitulating work by other authors; our novel contributions begin in Section 5.

Sets of Desirable Gamble Sets
Consider a finite possibility space X in which an uncertain variable X takes values. We denote by L (X ) the set of all gambles-real-valued functions-on X , often denoted by L when it is clear from the context what the possibility space is. We interpret a gamble f as an uncertain reward: if the actual outcome turns out to be x in X , then the agent's capital changes by f (x). For any two gambles f and g, we write f ≤ g when f (x) ≤ g(x) for all x in X , and we write f < g when f ≤ g and f = g. We identify a real constant α with the (constant) gamble that maps every element of X to α. We collect all the non-negative gambles-the gambles f for which f ≥ 0-in the set L (X ) ≥0 (often denoted by L ≥0 ), the positive ones-for which f > 0-in L (X ) >0 (often denoted by L >0 ), and the non-positive ones-for which f ≤ 0-in L (X ) ≤0 (often denoted by L ≤0 ).
We denote by Q(X ) the set of all finite but non-empty subsets of L (X ), also denoted by Q when it is clear from the context what X is. Q is a subset of the power set P(L ) of L . Elements of Q are the gamble sets.
De Bock and de Cooman [7] established a useful equivalent representation to choice functions, introduced in an imprecise-probabilistic context by Seidenfeld et al. [22]: Definition 1 (Set of desirable gamble sets) A set of desirable gamble sets K on X is a subset of Q(X ). We collect all the sets of desirable gamble sets in K := P(Q).
The idea is that the set of desirable gamble sets K collects all the gamble sets that contain at least one gamble that our agent strictly prefers over the status quo represented by 0, the gamble that will leave your capital unchanged whatever the outcome. A set of desirable gamble sets K is an equivalent representation to a choice function C: they are linked by K = {A ∈ Q : 0 / ∈ C({0} ∪ A)}. Therefore all our results will apply for choice functions as well. We will use sets of desirable gamble sets mainly for practical reasons: they are easier to work with.
De Bock and De Cooman [7] gave an axiomatization of coherent sets of desirable gamble sets-sets of desirable gamble sets of rational agents. We refer to their article for a justification of the axioms.
Definition 2 (Coherent set of desirable gamble sets) A set of desirable gamble sets K ⊆ Q is called coherent if for all A, A 1 and A 2 in Q, all {λ f,g , µ f,g : f ∈ A 1 , g ∈ A 2 } ⊆ R, and all f in L : We collect all the coherent sets of desirable gamble sets in the collection K (X ), often simply denoted by K .
Given two sets of desirable gamble sets K 1 and K 2 , we follow De Bock and De Cooman [7] in calling K 1 at most as informative as K 2 if K 1 ⊆ K 2 . The resulting partially ordered set (K , ⊆) is a complete lattice where intersection serves the role of infimum, and union that of supremum. De Bock and De Cooman [7,Theorem 8] furthermore show that the partially ordered set (K , ⊆) of coherent sets of desirable gamble sets is a complete meet-semilattice: given an arbitrary family {K i : i ∈ I} ⊆ K , its infimum inf{K i : i ∈ I} = i∈I K i is a coherent set of desirable gamble sets. This allows for conservative reasoning: it makes it possible to extend a partially specified set of desirable gamble sets to the most conservative-least informativecoherent one that includes it. This procedure is called natural extension: Definition 3 ([7, Definition 9]) For any assessment A ⊆ Q, we let K(A ) := {K ∈ K : A ⊆ K}. We call the as-1. We let N be the positive natural numbers. We let R >0 := {x ∈ R : x > 0} be the positive real numbers. sessment A consistent if K(A ) = / 0, and we then call cl K (A ) := K(A ) the natural extension of A .
One of the main results of De Bock and De Cooman [7] is their expression for the natural extension: If this is the case, then cl K (A ) = Rs(Posi(L s >0 ∪ A )). Here we used the set L s (X ) >0 := {{ f } : f ∈ L (X ) >0 }-often denoted simply by L s >0 -and the following two operations on K defined by Rs(K) := {A ∈ Q : (∃B ∈ K)B \ L ≤0 ⊆ A} and Binary choices can be modelled by a set of desirable gambles which collects all the gambles that the agent prefers to zero. They were introduced by Seidenfeld et al. [21], and have been studied extensively by Walley [24,25], De Cooman and Quaeghebeur [12], De Cooman and Miranda [11] and Quaeghebeur [18], amongst others. Formally, a set of desirable gambles is a subset D ⊆ L of gambles that are preferred over 0. We collect all the sets of desirable gambles in D := P(L ).
Definition 5 (Coherent set of desirable gambles) A set of desirable gambles D is called coherent if for all f and g in L , and λ and µ in R: We collect all the coherent sets of desirable gambles in D .
Just as we did for sets of desirable gamble sets, we call the set of desirable gambles D 1 at most as informative as set of desirable gambles Given a set of desirable gamble sets K, its binary behaviour is summarized in the set of desirable gambles Conversely, given a coherent set of desirable gambles D, there might be multiple coherent K that imply the same binary choices D K that are reflected in D: the non-empty collection {K ∈ K : D K = D} may have more than one element. However, it always contains one unique smallest element, which we call [23,Proposition 5]]. De Bock and De Cooman [8,Proposition 8] show that K D is coherent if and only if D is. We call any set of desirable gamble sets K binary if there is a set of desirable gambles D such that K = K D . The smallest coherent-called vacuous-set of desirable gamble sets is binary, and given by 0}. De Bock and De Cooman [7] establish an important representation result for coherent sets of desirable gamble sets. They show that any coherent set of desirable gamble sets K can be represented by a collection D of coherent sets of desirable gambles: 2 Theorem 7 (Representation [8, Theorem 9]) Any set of desirable gamble sets K is coherent if and only if there is a non-empty set D ⊆ D of coherent sets of desirable gambles such that K = {K D : D ∈ D}. We then say that D represents K. Moreover, K's largest representing set is D( Note that D(K) is an isotonic set: if D 1 ∈ D(K) and D 1 ⊆ D 2 , then D 2 ∈ D(K), for any D 1 and D 2 in D . admissible [16,19] and g > f}: 5 given any gamble set A, an option f is admissible precisely when there is some p in M such that f has highest p-expectation in A. The set of desirable gamble sets K M that corresponds to this, is given by K M is the set of desirable gamble sets that corresponds to the Eadmissible choice rule based on M . Let us show that K M is a coherent set of desirable gamble sets. One way to obtain this result is by checking that it satisfies all the rationality requirements from Definition 2, which is a cumbersome task. Thanks to Theorem 7 there is a much more elegant way to obtain this: we claim that K M is represented by the non-empty {D p : p ∈ M } ⊆ D , and is therefore coherent.
is the set of gambles that either have a positive p-expectation or are positive.
Jasper De Bock and Gert de Cooman showed us via private communication that Theorem 7 also allows for a simpler expression for the natural extension:

Conditioning
Suppose that we have a belief model about X, be it a coherent set of desirable gamble sets on X or a coherent set of desirable gambles on X , or-less generally-a set of probability mass functions on X. (We can think of a precise probability as a singleton.) When new information becomes available in the form of 'X assumes a value in some (non-empty) subset E of X', we can take this into account by conditioning our belief model on E.
We will let any event, except for the (trivially) impossible event / 0, serve as a conditioning event. We collect the allowed conditioning events in P / 0 (X ) := {E ⊆ X : E = / 0}. For any E in P / 0 (X ) and any gamble f on E, we let its multiplication I E f denote the gamble on X defined by for all x in X . I E f is the called-off version of f: if E does not occur, the gamble will yield 0.
Definition 10 (Conditioning) Given any set of desirable gamble sets K on X and any E in P / 0 (X ), we define the conditional set of desirable gamble sets K E on L (E) as K E := {A ∈ Q(E) : I E A ∈ K}, where for any A in Q(E) and E in P / 0 (X ), we let I E A := {I E g : g ∈ A} be a set of called-off gambles.
It follows at once that conditioning preserves the order: if This definition coincides with the usual definition for sets of desirable gambles, in the sense that : I E f ∈ D} is the set of desirable gambles conditional on E. In order to elegantly work with K E's representation in terms of sets of desirable gambles, let us define D E := {D E : D ∈ D} for any D ⊆ D.

Proposition 11 ([23, Propositions 7 and 8])
Consider any set of desirable gamble sets K on X and any Example 2 Let us build on Example 1, and condition the E-admissible set of desirable gamble sets K M on an So the conditional set of desirable gamble sets K M G is equal to the E-admissible set of desirable gamble sets K {p|G : p∈M } obtained by an element-wise application of Bayes's rule on M .

Multivariate Sets of Desirable Gamble Sets
In this section, we will generalize the multivariate study of desirability by De Cooman and Miranda [11] to choice models. We will provide the linear space of gambles, on which we define our sets of desirable gamble sets, with a more complex structure: we will consider the vector space of all gambles whose domain is a Cartesian product of a finite number of finite possibility spaces. More specifically, consider n in N variables X 1 , . . . , X n that assume values in the finite possibility spaces X 1 , . . . , X n , respectively. Belief models about these variables X 1 , . . . , X n will be defined using gambles on X 1 , . . . , X n . We also consider gambles on the Cartesian product × n k=1 X k , giving rise to the ∏ n k=1 |X k |-dimensional linear space L ( × n k=1 X k ).

Basic Notation & Cylindrical Extension
For every nonempty subset I ⊆ {1, . . . , n} of indices, we let X I be the tuple of variables that takes values in X I := × r∈I X r . We will denote generic elements of X I as x I or z I , whose components are x i := x I (i) and z i := z I (i), for all i in I. As we did before, when I = {k, . . . , } for some k, in {1, . . . , n} with k ≤ , we will use as a shorthand notation X k: := (X k , . . . , X ), taking values in X k: and whose generic elements are denoted by x k: := (x k , . . . , x ).
We assume that the variables X 1 , . . . , X n are logically independent, meaning that for each non-empty subset I of {1, . . . , n}, x I may assume every value in X I .
It will be useful for any gamble f on X 1:n , any nonempty proper subset I of {1, . . . , n} and any x I in X I , to interpret the partial map f(x I , · ) as a gamble on X I c , where I c := {1, . . . , n} \ I. We will need a way to relate gambles on different domains: Definition 12 (Cylindrical extension) Given two disjoint and non-empty subsets I and I of {1, . . . , n} and any gamble f on X I , we let its cylindrical extension f * to X I∪I be defined by f * (x I , x I ) := f(x I ) for all x I in X I and x I in X I . Similarly, given any set of gambles A ⊆ L (X I ), we let its cylindrical extension A * ⊆ L (X I∪I ) be defined as Formally, f * belongs to L (X I∪I ) while f belongs to L (X I ). However, f * is completely determined by f and vice versa: they clearly only depend on the value of X I , and as such, they contain the same information and correspond to the same transaction. They are therefore indistinguishable from a behavioural point of view.
Remark 13 As in [9,11], we will frequently use the simplifying device of identifying a gamble f on X I with its cylindrical extension f * on X I∪I , for any disjoint and non-empty subsets I and I of the index set {1, . . . , n}. This convention allows us, for instance, to identify L (X I ) with a subset of L (X 1:n ), and, as another example, for any set A ⊆ L (X 1:n ), to regard A ∩ L (X I ) as those gambles in A that depend on the value of X I only. Therefore, for any event E in P / 0 (X I ) we can identify the gamble I E with I E×X I c , and hence also the event E with E × X I c .
Marginalization Suppose we have a set of desirable gamble sets K on X 1:n modelling an agent's beliefs about the variable X 1:n . What is the information that K contains about X O , where O is some non-empty subset of the index set {1, . . . , n}? Marginalization captures this information.
Definition 14 (Marginalization) Given any non-empty subset O of {1, . . . , n} and any set of desirable gamble sets K on X 1:n , its marginal set of desirable gamble sets marg O K on X O is defined as It follows at once from Definition 14 that marginalization preserves the order: if This definition coincides with the usual definition for sets of desirable gambles, in the sense that Conditioning on Variables In Section 3 we have seen how we can condition sets of desirable gamble sets on events. Here, we take a closer look at conditioning in a multivariate context.
Suppose we have a set of desirable gamble sets K on X 1:n , representing an agent's beliefs about the value of X 1:n . Assume now that we obtain the information that the I-tuple of variables X I -where I is a non-empty subset of {1, . . . , n}-assumes a value in a certain non-empty subset E I of X I . There is no new information about the other variables X I c . How can we condition K on this new information?
This is a particular instance of Definition 10, with the following specifications: X = X 1:n and E = E I × X I c . The indicator I E of the conditioning event E satisfies I E (x 1:n ) = I E I (x I ) for all x 1:n in X 1:n , and taking Remark 13 into account, therefore I E = I E I . Equation (1) defines the multiplication of a gamble f on E I × X I c with I E I to be a gamble I E I f on X 1:n , given by, for all x 1:n in X 1:n : and the multiplication of I E I with a set A of gambles on E I × X I c is the set I E I A = {I E I f : f ∈ A} of gambles on X 1:n . Now that we have instantiated all the relevant aspects of Definition 10, we see that K E I = {A ∈ Q(E I × X I c ) : I E I A ∈ K}. Proposition 11 guarantees that K E I is represented by The conditional set of desirable gamble sets K E I is defined on gambles on E I × X I c . However, usuallysee, for instance, [6,11]-conditioning on information about X I results in a model on X I c . We therefore consider marg I c (K E I ) = {A ∈ Q(X I c ) : I E I A ∈ K} as the set of desirable gamble sets that represents the conditional beliefs about X I c , given that X I ∈ E I . Proposition 11 guarantees the coherence of marg I c (K E I ), for any coherent K.

Independent Natural Extension
Now that the basic operations of multivariate sets of desirable gamble sets-marginalization and conditioning-are in place, we are ready to look at a simple type of structural assessment. The assessment that we will consider, is that of epistemic independence, which we define to be a symmetrized version of epistemic irrelevance.

Definition 16 (Epistemic (subset) irrelevance)
Consider any disjoint and non-empty subsets I and O of {1, . . . , n}. We call X I epistemically (subset) irrelevant to X O when learning about the value of X I does not influence or change the agent's beliefs about X O . A set of desirable gamble sets K on X 1:n is said to satisfy epistemic subset irrelevance of X I to X O when The idea behind this definition is that observing that X I belongs to E I turns K into the conditioned set of desirable gamble sets K E I on E I × X I c . Then requiring that learning that X I belongs to E I does not affect the agent's beliefs about X O amounts to requiring that the marginal models of K and K E I be equal. This definition is a generalization of De Cooman and Miranda [11]'s definition for sets of desirable gambles. Besides their use of the less expressive models of sets of desirable gambles, there is another difference: De Cooman and Miranda [11] consider epistemic value irrelevance, which requires the analogue of Equation (3) only for events of the form E I = {x I }, with x I ∈ X I . De Bock [6,Example 2] shows that the two notions do indeed come apart: he gives a coherent set of desirable gambles that satisfies epistemic value irrelevance of X 1 to X 2 , but not epistemic subset irrelevance. Given the connection between sets of desirable gambles and sets of desirable gamble sets, this example establishes that the two notions come apart also in the context of sets of desirable gamble sets. We follow De Bock [6] in considering epistemic subset-irrelevance to be the more natural of the two irrelevance concepts, as it requires all information about the value of X I to be irrelevant, including partial information of the form X I ∈ E I , and not only of the form X I = x I .

Definition 17 (Epistemic (subset) independence)
We call X 1 , . . . X n epistemically (subset) independent when learning about the values of any of them does not influence or change the agent's beliefs about the remaining ones: for any two disjoint non-empty subsets I and O of {1, . . . , n}, X I is epistemically subset irrelevant to X O . We call a set of desirable gamble sets K on X 1:n epistemically (subset) independent when for all disjoint non-empty subsets I and O of {1, . . . , n}.
Independence assessments are useful in constructing joint sets of desirable gamble sets from local ones. Suppose we have a coherent set K of desirable gamble sets on X , for each in {1, . . . , n}, and an assessment that the variables X 1 , . . . , X n are epistemically subset independent. Then how can we combine the coherent local assessments K and this structural independence assessment into a coherent set of desirable gamble sets on X 1:n in a way that is as conservative as possible? If we call any coherent and epistemically independent K on X 1:n that marginalizes to K for all in {1, . . . , n} an independent product of K 1 , . . . K n , this means we are looking for the smallest independent product, which we will call the independent natural extension of K 1 , . . . The set A 1:n\{ j}→{ j} expresses all the epistemic subset independence assessments from Equation (4), which the independent natural extension must satisfy. This is a difference with the project of [11]: the original theorem [11,Theorem 19] considers the subset A 1:n\{ j}→{ j} := {I {x} f : f ∈ D j and x ∈ X 1:n\{ j} }, 6 expressing the epistemic value independence assessments. As expected, A 1:n\{ j}→{ j} ⊂ A 1:n\{ j}→{ j} , which is consistent with the fact that epistemic value independence is a weaker requirement than epistemic subset independence, but posi(A 1:n\{ j}→{ j} ) = posi(A 1:n\{ j}→{ j} ), 7 resulting in equal independent natural extensions: the smallest coherent set of desirable gambles that is epistemically value independent also is epistemically subset independent. As we will see in Theorem 20, the independent natural extension for sets of desirable gamble sets will be represented by a collection of independent natural extensions for sets of desirable gambles, and therefore the smallest independent product for sets of desirable gamble sets will also coincide for these two notions of independence. By default, we will just say 'epistemic irrelevance' and 'epistemic independence' to indicate their subset variants.
We will show that the independent natural extensionthe smallest independent product-of K 1 , . . . , K n is exactly n j=1 K j . Before we prove this, it will be useful to show the following connection between the assessments A 1:n\{ j}→{ j} and A 1:n\{ j}→{ j} for the two types of belief models:

Proposition 19
Consider, for each j in {1, . . . , n}, a coherent set of desirable gambles D j on X j . Then K n j=1 D j = n j=1 K D j , so K n j=1 D j is the smallest element of K that includes n j=1 {I E A : A ∈ K D j and E ∈ P / 0 (X 1:n\{ j} )}.
Theorem 20 (Independent natural extension) Consider, for each j in {1, . . . , n}, a coherent set of desirable gamble sets K j on X j . Then the smallest independent product of K 1 , . . . , K n is given by n j=1 K j . Furthermore, n j=1 K j is represented by Theorem 20 establishes that the independent natural extension n j=1 K j can be calculated using the independent natural extension for sets of desirable gambles, because n j=1 D(K j ) is a representation of n j=1 K j . So n j=1 K j is represented by a collection n j=1 D(K j ) of independent natural extensions of sets of locally desirable gambles. However, this does not also imply that the largest representation D( n j=1 K j ) of n j=1 K j will only consist of such sets of desirable gambles. Since D( n j=1 K j ) is an isotonic set, it will include some maximal 8 set of desirable gamblesD, because every coherent set of desirable gambles is dominated by a maximal one. 9 But De Cooman and Miranda [11,Proposition 23] have established that any maximal set of desirable gambles on X 1:n is not an independent natural extension of local sets of desirable gambles on X 1 , . . . , X n when n = |X 1 | = |X 2 | = 2. They leave this question open for cases where max{n, |X k | : k ∈ {1, . . . , n}} ≥ 2. This means that D( n j=1 K j ) can indeed contain sets of desirable gambles that are not the result of an independent natural extension, and, if the answer to their open question is 'yes', will always contain such sets of desirable gambles.
A useful property of the independent natural extension n j=1 D j for sets of desirable gambles is that it is associative [11,Theorem 20]: A maximal set of desirable gambles is an undominated element of the complete infimum-semilattice (D , ⊆): they are the maximally informative coherent sets of desirable gambles. They are characterized by the property f ∈D or − f ∈D, for any non-zero f in L . We refer to [2,12] for a proof of this representation and more information. 9. See Couso and Moral [2,Theorem 20] and De Cooman and Quaeghebeur [12, Theorem 3] for a proof of this statement.
2 ∈L 2 D 2 for any disjoint non-empty subsets L 1 and L 2 of {1, . . . , n}. This property allows for a modular construction of the independent natural extension. It implies that our independent natural extension n j=1 K j is associative too: Proposition 21 (Associativity) Consider, for each j in {1, . . . , n} a coherent set of desirable gamble sets K j on X j , and let L 1 and L 2 be any disjoint non-empty subsets of {1, . . . , n}. Then Note that Proposition 21 implies a stronger marginalization property, namely marg O n j=1 K j = j∈O K j for any non-empty subset O of {1, . . . , n}. Indeed, by considering L 1 := O and L 2 := {1, . . . , n} \ L 1 we find that n j=1 K j = j∈O K j ⊗ j∈L 2 K j , so it is the independent natural extension of two sets of desirable gamble sets j∈O K j and j∈L 2 K j . By Theorem 20 n j=1 K j then indeed marginalizes to j∈O K j .
A Stronger Independence Requirement In Definition 17, we defined a set of desirable gamble sets K to be epistemically independent when Equation (4) holds, or, in other words, when for all disjoint non-empty subsets I and O of {1, . . . , n}, E in P / 0 (X I ) and A in Q(X O ). This requires that, if A is a desirable gamble set, then at least one gamble of {I E f : f ∈ A} = I E A should be preferred to zero, and hence I E A ∈ K. But one might argue that independence-or indeed, also irrelevance-should require something stronger, namely that should belong to K, for every choice of conditioning events E A := {E f : f ∈ A} ⊆ P / 0 (X I ). The gamble set E A ·A is the result of multiplying any gamble f in A with its corresponding indicator I E f with E f ∈ E A , so E A · A contains multiplications with (indicators of) different events, rather than then same event E. This leads to the following stronger independence requirement, which, as we shall see, has a tight connection with epistemic independence: We say that K satisfies the stronger notion of independence if for all disjoint non-empty subsets I and O of {1, . . . , n}, E A ⊆ P / 0 (X I ) and A in Q(X O ). This is a valid generalization of independence in the sense that, for any epistemically independent set of desirable gambles D, the binary K D satisfies the requirement in Equation (9), too. We this independence notion at least as compelling as epistemic independence. As E A may contain only one event E, in which case E A · A = {I E f : f ∈ A} = I E A, this requirement implies the usual requirement of Equation (7). However, the stronger requirement Equation (9) is satisfied by the independent natural extension:

Proposition 22
The independent natural extension n j=1 K j of K 1 , . . . , K n satisfies the stronger requirement of Equation (9). As a consequence, any independent product of K 1 , . . . , K j includes n j=1 E A · A : A ∈ K j and E A ⊆ P / 0 (X 1:n\{ j} ) .
Let us consider the E-admissible set of desirable gamble sets K M , defined in Example 1. Since M marginalizes to M 1 , . . . , M n and satisfies independence, the set of desirable gamble sets K M , too, will marginalize to M 1 , . . . , M n and satisfy independence: K M is an independent product of K M 1 , . . . , K M n .
To see that it indeed does satisfy the alternative independence requirement of Equation (9), consider any A in Q(X O ) and G A ⊆ P / 0 (X I ). Infer for any p in M , f in A and G f in

Contrasting Epistemic Irrelevance with S-Irrelevance
We have chosen to investigate the independent natural extension of sets of desirable gamble sets according to the standard that we have called 'epistemic irrelevance', but there are numerous other notions of irrelevance we might have investigated. One particularly interesting conception of irrelevance is a notion due to Teddy Seidenfeld [22,Section 4] and recently investigated by Jasper De Bock and Gert de Cooman [1]. The basic idea is that one proposition is irrelevant to another if the agent doesn't regard learning about the first proposition as valuable to decisions that depend only on whether the second proposition is true [1, Section 4.1]: "When two variables, X and Y , are 'independent' then it is not reasonable to spend resources in order to use the observed value of one of them, say X, to choose between options that depend solely on the value of the other variable, Y ." To translate this into a workable definition, consider any partition P of X's finite possibility space X , and for every element E of P, a gamble f E on Y 's finite possibility space Y .
Then the suggested notion of irrelevance, which De Bock 10. We use the (topological) interior int(Σ X j ) of Σ X j to make sure that every outcome in X j has a (strictly) positive probability for every element of M j . and De Cooman [1] term S-irrelevance, 11 is that an agent who judges that X is S-irrelevant to Y will be forced to disprefer the composite gamble ∑ E∈P I E (X) f E (Y ) − ε, which is the result of paying ε to find out which E * in P occursthe E * such that X ∈ E * -in order to decide to take the gamble f E * (Y ), to at least one of { f E : E ∈ P}. In other words, for all E in P, and ε ∈ R >0 . X and Y are called S-independent when X is S-irrelevant to Y and vice versa.
When X is a binary variable, meaning that X = {x 1 , x 2 }, this requirement reduces to S-irrelevance is an intuitively very compelling standard, which raises a natural question: how is our concept of epistemic irrelevance related? It is already clear from De Bock and De Cooman [1]'s analysis of S-irrelevance that it is not entailed by our notion of epistemic irrelevance; as they note [1,Corollary 1], under suitable continuity conditions 12 S-irrelevance has the surprising consequence of being mixing, which loosely speaking implies that the set of desirable gamble sets is represented by a collection of linear prevision. Let us give an explicit example, specialized to our context.

Example 4
Consider the independent natural extension K 1 ⊗ K 2 of two vacuous local models K 1 and K 2 on X and Y , respectively. We will show that this is the vacuous K v on X × Y . Indeed, K v marginalizes to K 1 and K 2 , and it also satisfies the independence requirements of Equation (7): for any A in Q(X ) and E in P / 0 (Y ), or A in Q(Y ) and E in P / 0 (X ). So K v is an independent product. Since it is the smallest coherent set of desirable gamble sets, it is equal to Let us show that K 1 ⊗ K 2 does not satisfy S-irrelevance, and therefore also not S-independence. To this end, assume that both X = {x 1 , x 2 } and Y = {y 1 , y 2 } are binary.
As far as we know, however, whether S-irrelevance entails our standard of epistemic irrelevance has not yet been shown. In the following section, we develop an example which shows that it is possible to satisfy S-irrelevance while flouting (both value and subset) epistemic irrelevance. Thus, neither S-nor epistemic irrelevance entails the other.
Violating Epistemic Irrelevance While Satisfying S-Irrelevance There are two general ways we suspect it is possible to violate epistemic subset irrelevance while satisfying S-irrelevance. A variable X fails to be epistemically irrelevant to a variable Y just in case A ∈ K but I E A / ∈ K, or I E A ∈ K but A / ∈ K, for some E in P / 0 (X ) and A in Q(Y ). In essence, there is some proposition about the value of X that the agent can learn which will change their views about the preferences for some gambles that depend only on Y . The question we are interested in is whether there is a way for an agent who knows that learning E will change their views about the preferences between these gambles to not place any real monetary value on learning it. There are at least two ways that it occurs to us that this could happen: • the agent thinks that there is no real value gained by using the informed strategy over merely accepting a wager without learning; • the agent is certain that the experiment they are (not) paying for will not yield the outcome which would change their desires.
In the present paper, we leave the latter kind of case as a conjecture. In this section, we develop an example of the former. The framework of sets of desirable gamble sets (and indeed, the less expressive framework of sets of desirable gambles) is capable of representing an agent as believing that one outcome is infinitesimally more likely than another. In the multivariate arena, this raises the possibility of correlations that generate only infinitesimal change in belief. Information that generates such an infinitesimal change will not have any real expected value as long as the gambles that are at issue have only finite value, which is consistent with S-irrelevance; Nonetheless, learning the information does make an identifiable change to which gambles the agent finds desirable, and thus subset irrelevance is violated.
Example 5 A factory produces two kinds of coins: coins that are fair (heads and tails are equally likely) and coins that are infinitesimally biased in favor of heads (heads is more likely than tails but not by any definite amount). Consider an agent who knows that a coin produced by this factory is about to be flipped; the above description of the factory is all they know. They have no beliefs about the proportion of coins of each type the factory produces, or any specific reason to believe that the coin in question is of one type or the other.
The agent is offered the following decision problem: they can accept a wager that pays some real payout a if the coin lands heads and −a if tails, they can decline the wager (maintain the status quo, accept the zero gamble), or they can pay some small (real-valued) fee ε > 0 to learn the type of the coin, and then decide whether to accept or reject the (a, −a) wager. Let X take values in X := {F,U} representing whether the coin is fair or unfair and let Y take values in Y := {H, T } representing whether the coin lands heads or tails. For ease of future reference, we denote Lemma 24 D is a coherent set of desirable gambles. Moreover, it is no independent product, and it therefore fails epistemic irrelevance from X to Y .
So D-and therefore K D , which is coherent since D isfails to satisfy epistemic value irrelevance, and therefore also epistemic subset irrelevance. However, despite this difference in desirability dependent on learning about X, there is no positive value ε that the agent will be willing to pay to learn the bias before deciding whether to accept or reject a gamble on the outcome of the flip, so a gamble on Y . More generally, there are no gambles f and g on Y such that the agent would pay to learn the bias of the coin before deciding between f and g.
To show this formally, we consider K D and show that it satisfies S-irrelevance: Lemma 25 X is S-irrelevant to Y with respect to K D .
Thus, we have a case where X is not epistemically irrelevant to Y , but X is S-irrelevant to Y . The upshot is that there are cases where learning about a variable X makes an identifiable change to which gambles, defined only on Y , an agent prefers, but the agent sees this difference as negligible in (real) value.

Discussion
Independence is an interesting concept. When we model uncertainty with precise probabilities, it seems univocal. But when we model uncertainty with imprecise probabilities, it fractures into a multiplicity of distinct concepts, including: • complete independence for sets of probabilities [3,20]; • independence in selection for lower previsions [10]; • strong independence for lower previsions and sets of desirable gambles [11]; • epistemic independence (value and subset) for sets of desirable gambles [17]; • epistemic h-independence for lower previsions and credal sets [6]; • S-independence for choice functions [1]; These concepts collapse in the limit, when applied to precise probabilities, but come apart in general.
Independence is also an important concept. For example, many have thought that when pooling expert opinions we ought to preserve unanimous judgments of independence [14,15,13]. Take another example: causal modelling. Causal Bayesian networks consist of a directed acyclic graph together with an appropriate probability distribution. They are popular formal tools for modelling causal relationships. Independence judgments play a key role in constructing causal Bayes nets. Missing edges between variables in the graph of a causal Bayes net indicate that those variables are causally independent of one another.
In this paper we investigated epistemic independence in the general framework of sets of desirable gamble sets. Our results indicate that the independent natural extension can be calculated by performing the eponymous operation on a representing collection of sets of desirable gambles, but this collection may be infinite.
In addition, we took some initial steps to compare epistemic independence with another attractive notion of independence proposed by Teddy Seidenfeld: S-independence. Recently, Jasper De Bock and Gert de Cooman [1, Corollary 1] showed that if an Archimedean set of desirable gamble sets renders a variable X "credibly indeterminate", then judging that X is S-irrelevant to Y forces you to choose between gambles on Y using E-admissibility. Judgments of S-irrelevance, it turns out, are much more informative than they appear at first glance.
There are still a number of open questions about epistemic independence for sets of desirable gamble sets. For example, Cozman and Seidenfeld [5] explore the notion of layer independence for full conditional probability measures. Cozman [4] shows that the only extant concept of independence for (non-convex) sets of probabilities that has a range of desirable graphoid properties is element-wise layer independence. It is an open question what additional structural constraints on coherent sets of desirable gamble sets are necessary and sufficient to secure the relevant graphoid properties.
As sets of desirable gamble sets generalize many of the extant imprecise-probabilistic uncertainty models, including sets of desirable gambles, lower previsions, and sets of probability mass functions, they may be expressive enough to unify some of the aforementioned independence concepts. Example 3 is one instance of this, where we show that an E-admissible set of desirable gamble sets based on a completely independent credal set is an (epistemically) independent product. We suspect Proposition 22 to be helpful in the connection with some of the other independence concepts. We intend to investigate these connections, with the hope to obtain a unifying theory.