4.3 Independence for Plausibility Measures

Definition 4.2.2 can be easily adapted to each of the notions of conditioning discussed in Chapter 3. It is perhaps easiest to study independence generally by considering it in the context of plausibility—all the other notions are just special cases.

Definition 4.3.1

Given this definition of conditional independence for plausibility, it is then possible to ask whether the obvious analogues of CI1[μ]–CI5[μ] hold. Let CIn[Pl] be the result of replacing the probability measure μ by the plausibility measure Pl in CIn[μ]. CI1[Pl], CI2[Pl], and CI5[Pl] are easily seen to hold for all cpms Pl (Exercise 4.6), but CI3[Pl] and CI4[Pl] do not hold in general. CI3 and CI4 do hold for conditional ranking and for Pl_P, but it is easy to see that they do not hold for conditional possibility or for conditional belief, no matter which definition of conditioning is used (Exercise 4.7).

How critical is this? Consider CI3. If U is independent of V, should it necessarily also be independent of V? Put another way, if U is unrelated to V, should it necessarily be unrelated to V as well? My own intuitions regarding relatedness are not strong enough to say definitively. In any case, if this seems like an important component of the notion of independence, then the definition can easily be modified to enforce it, just as the current definition enforces symmetry.

Other notions of independence have been studied in the literature for specific representations of uncertainty. There is a general approach called noninteractivity, which takes as its point of departure the observation that μ(U ∩ V) = μ(U) μ(V) if U and V are independent with respect to the probability measure μ (cf. Proposition 4.1.2). While noninteractivity was originally defined in the context of possibility measures, it makes sense for any algebraic cpm.

Definition 4.3.2

U and V do not interact given V′ (with respect to the algebraic cpm Pl), denoted NI_Pl(U, V | V′), if V′ ∊ ′ implies that Pl(U ∩ V | V′) = Pl(U | V′) ⊗ Pl(V | V′).

Proposition 4.2.3 shows that, for conditional probability defined from unconditional probability, noninteractivity and independence coincide. However, they do not coincide in general (indeed, Example 4.1.4 shows that they do not coincide in general even for conditional probability spaces). In general, independence implies noninteractivity for algebraic cps's.

Lemma 4.3.3

If (W, , ′, Pl) is an algebraic cps, then I_Pl(U, V | U′) implies NI_Pl(U, V | U′).

Proof See Exercise 4.8.

Noninteractivity and independence do not necessarily coincide in algebraic cps's, as the following example shows:

Example 4.3.4

Suppose W ={w₁, w₂, w₃, w₄}, Poss(w₁) = Poss(w₂) = Poss(w₃) = 1/2, and Poss(w₄) = 1. Let U ={w₁, w₂} and V ={w₂, w₃}. Then NI_Poss(U, V | W), since Poss(U | W) = 1/2, Poss(V | W) = 1/2, and Poss(U ∩ V | W) = 1/2 (recall that ⊗ is min for possibility measures). But Poss(V | U) = 1 ≠ Poss(V), so it is not the case that I_Poss(U, V | W).

So what is required for noninteractivity to imply independence? It turns out that Alg4′ (as defined in Section 3.9) suffices for standard algebraic cps's. (Recall that a cps (W, , ′, Pl) is standard if ′ ={U : {Pl(U) ≠ ⊥}}.)

Lemma 4.3.5

If (W, , ′, Pl) is a standard algebraic cps that satisfies Alg4′, then NI_Pl(U, V | U′) implies I_Pl(U, V | U′).

Proof See Exercise 4.9.

It is easy to see that the assumption of standardness is necessary in Lemma 4.3.5 (Exercise 4.10). For a concrete instance of this phenomenon, consider the cps implicitly defined in Example 3.2.4. This cps is algebraic (Exercise 4.11) but nonstandard, since conditioning on {w₂} is allowed although μ^s(w₂) = 0. Example 4.1.4 shows that, in this cps, noninteractivity does not imply independence for U ={w₁, w₃} and V ={w₂, w₃}.

The fact that noninteractivity and conditional independence coincide for the conditional plausibility spaces constructed from unconditional probability measures and ranking functions now follows from Lemmas 4.3.3 and 4.3.5. Since neither Poss(U | V) nor Pl_P satisfy Alg4′, it is perhaps not surprising that in neither case does noninteractivity imply conditional independence. Example 4.3.4 shows that is the case for Poss(V | U). The following example shows this for Pl_퓹:

Example 4.3.6

Suppose that a coin is known to be either double-headed or double-tailed and is tossed twice. This can be represented by 퓹 = {μ₀, μ₁} where μ₀(hh) = 1 and μ₀(ht) = μ₀(th) = μ₀(tt) = 0, while μ₁(tt) = 1 and μ₁(ht) = μ₁(th) = μ₁(hh) = 0. Let H¹ ={hh, ht} be the event that the first coin toss lands heads, and let H² = {hh, th} be the event that the second coin toss lands heads. Clearly there is a functional dependence between H¹ and H². Intuitively, they are related and not independent. On the other hand, it is easy to check that H¹ and H² are independent with respect to both μ₀ and μ₁ (since μ₀(H¹) = μ₀(H²) = μ₀(H¹ ∩ H²) = 1 and μ₁(H¹) = μ₁(H²) = μ₁(H¹ ∩ H²) = 0). It thus follows that H¹ and H² do not interact: NI_{Pl_퓹} (H¹, H²) holds, since Pl_퓹(H¹ ∩ H²) = Pl_퓹 (H¹) = Pl_퓹(H²) = (1, 0). On the other hand, I_{Pl_퓹} (H¹, H²) does not hold. For example, f_H¹(μ₁) = 0 while f_H¹|H²(μ₁) = ⋆. (See the notes to this chapter for more discussion of this example.)

Yet other notions of independence, besides analogues of Definition 4.3.1 and noninteractivity, have been studied in the literature, particularly in the context of possibility measures and sets of probability measures (see the notes for references). While it is beyond the scope of this book to go into details, it is worth comparing the notion of independence for Pl_퓹 with what is perhaps the most obvious definition of independence with respect to a set 퓹 of probability measures—namely, that U and V are independent if U and V are independent with respect to each μ ∊ 퓹. It is easy to check that I_{Pl_퓹} (U, V | V′) implies I_μ(U, V | V′) for all μ ∊ 퓹 (Exercise 4.13), but the converse does not hold in general, as Example 4.3.6 shows.

This discussion illustrates an important advantage of thinking in terms of notions of uncertainty other than probability. It forces us to clarify our intuitions regarding important notions such as independence.