2.5 Possibility Measures

Possibility measures are yet another approach to assigning numbers to sets. They are based on ideas of fuzzy logic. Suppose for simplicity that W, the set of worlds, is finite and that all sets are measurable. A possibility measure Poss associates with each subset of W a number in [0, 1] and satisfies the following three properties:

Poss1. Poss(∅) = 0.
Poss2. Poss(W) = 1.
Poss3. Poss(U ∪ V) = max(Poss(U), Poss(V)) if U and V are disjoint.

The only difference between probability and possibility is that if A and B are disjoint sets, then Poss(U ∪ V) is the maximum of Poss(U) and Poss(V), while μ(U ∪ V) is the sum of μ(U) and μ(V). It is easy to see that Poss3 holds even if U and V are not disjoint (Exercise 2.31). By way of contrast, P2 does not hold if U and V are not disjoint.

It follows that, like probability, if W is finite and all sets are measurable, then a possibility measure can be characterized by its behavior on singleton sets; Poss(U) = max_u∈U Poss(u). For Poss2 to be true, it must be the case that max_{w ∈ W} Poss(w) = 1; that is, at least one element in W must have maximum possibility.

Also like probability, without further assumptions, a possibility measure cannot be characterized by its behavior on singletons if W is infinite. Moreover, in infinite spaces, Poss1–3 can hold without there being any world w ∈ W with Poss(w) = 1, as the following example shows:

Example 2.5.1

Consider the possibility measure Poss₀ on 풩 such that Poss₀(U) = 0 if U is finite, and Poss(U) = 1if U is infinite. It is easy to check that Poss₀ satisfies Poss1–3, even though Poss₀(n) = 0 for all n ∈ 풩 (Exercise 2.32(a)).

Poss3 is the analogue of finite additivity. The analogue of countable additivity is
Poss3′. Poss(∪^∞_i=1U_i) = sup^∞_i=1 Poss(U_i) if U₁, U₂, … are pairwise disjoint sets.

It is easy to see that Poss₀ does not satisfy Poss3′ (Exercise 2.32(b)). Indeed, if W is countable and Poss satisfies Poss1 and Poss3′, then it is immediate that Poss(W) = 1 = sup_w∈W Poss(w). Thus, there must be worlds in W with possibility arbitrarily close to 1. However, there may be no world in W with possibility 1.

Example 2.5.2

If Poss₁ is defined on 풩 by taking Poss₁(U) = sup_{n ∈ U} (1 − 1/n), then it satisfies Poss1, Poss2, and Poss3′, although clearly there is no element w ∈ W such that Poss(w) = 1 (Exercise 2.32(c)).

If W is uncountable, then even if Poss satisfies Poss1, Poss2, and Poss3′, it is consistent that all worlds in W have possibility 0. Moreover, Poss1, Poss2, and Poss3′ do not suffice to ensure that the behavior of a possibility measure on singletons determines its behavior on all sets.

Example 2.5.3

Let Poss₂ be the variant of Poss₀ defined on by taking Poss₂(U) = 0 if U is countable and Poss₂(U) = 1if U is uncountable. Then Poss₂ satisfies Poss1, Poss2, and Poss3′, even though Poss₂(w) = 0 for all w ∈ W (Exercise 2.32(d)). Now let Poss₃ be defined on by taking

Then Poss₃ is a possibility measure that satisfies Poss1, Poss2, and Poss3′ (Exercise 2.32(e)). Clearly Poss₂ and Poss₃ agree on all singletons (Poss₂(w) = Poss₃(w) = 0 for all w ∈ ), but Poss₂([0, 1/2]) = 1 while Poss₃([0, 1/2]) = 1/2, so Poss₂ ≠ Poss₃.

To ensure that a possibility measure is determined by its behavior on singletons, Poss3′ is typically strengthened further so that it applies to arbitrary collections of sets, not just to countable collections:

Poss3⁺. For all index sets I, if the sets U_i, i ∊ I, are pairwise disjoint, then Poss (∪_{i ∊ I}) = sup_{i ∊ I} Poss(U_i).

Poss1 and Poss3⁺ together clearly imply that there must be elements in W of possibility arbitrarily close to 1, no matter what the cardinality of W. Moreover, since every set is the union of its elements, a possibility measure that satisfies Poss3⁺ is characterized by its behavior on singletons; that is, if two possibility measures satisfying Poss3⁺ agree on singletons, then they must agree on all sets.

Both Poss3′ and Poss3⁺ are equivalent to continuity properties in the presence of Poss3. See Exercise 2.33 for more discussion of these properties.

It can be shown that a possibility measure is a plausibility function, since it must satisfy (2.16) (Exercise 2.34). The dual of possibility, called necessity, is defined in the obvious way:

Of course, since Poss is a plausibility function, it must be the case that Nec is the corresponding belief function. Thus, Nec(U) ≤ Poss(U). It is also straightforward to show this directly from Poss1–3 (Exercise 2.35).

There is an elegant characterization of possibility measures in terms of mass functions, at least in finite spaces. Define a mass function m to be consonant if it assigns positive mass only to an increasing sequence of sets. More precisely, m is a consonant mass function if m(U) > 0 and m(U′)> 0 implies that either U ⊆ U′ or U′ ⊆ U. The following theorem shows that possibility measures are the plausibility functions that correspond to a consonant mass function:

Theorem 2.5.4

If m is a consonant mass function on a finite space W, then Plaus_m, the plausibility function corresponding to m, is a possibility measure. Conversely, given a possibility measure Poss on W, there is a consonant mass function m such that Poss is the plausibility function corresponding to m.

Proof See Exercise 2.36.

Theorem 2.5.4, like Theorem 2.4.3, depends on W being finite. If W is infinite, it is still true that if m is a consonant mass function on W, then Plaus_m is a possibility measure (Exercise 2.36). However, there are possibility measures on infinite spaces that, when viewed as plausibility functions, do not correspond to a mass function at all, let alone a consonant mass function (Exercise 2.37).

Although possibility measures can be understood in terms of the Dempster-Shafer approach, this is perhaps not the best way of thinking about them. Why restrict to belief functions that have consonant mass functions, for example? Many other interpretations of possibility measures have been provided, for example, in terms of degree of surprise (see the next section) and betting behavior. Perhaps the most common interpretation given to possibility and necessity is that they capture, not a degree of likelihood, but a (subjective) degree of uncertainty regarding the truth of a statement. This is viewed as being particularly appropriate for vague statements such as "John is tall." Two issues must be considered when deciding on the degree of uncertainty appropriate for such a statement. First, there might be uncertainty about John's actual height. But even if an agent knows that John is 1.78 meters (about 5 foot 10 inches) tall, he might still be uncertain about the truth of the statement "John is tall." To what extent should 1.78 meters count as tall? Putting the two sources of uncertainty together, the agent might decide that he believes the statement to be true to degree at least .3 and at most .7. In this case, the agent can take the necessity of the statement to be .3 and its possibility to be .7.

Possibility measures have an important computational advantage over probability: they are compositional. If μ is a probability measure, given μ(U) and μ(V), all that can be said is that μ(U ∪ V) is at least max(μ(U), μ(V)) and at most min(μ(U) + μ(V), 1). These, in fact, are the best bounds for μ(U ∪ V) in terms of μ(U) and μ(V) (Exercise 2.38). On the other hand, as Exercise 2.31 shows, Poss(U ∪ V) is determined by Poss(U) and Poss(V): it is just the maximum of the two.

Of course, the question remains as to why max is the appropriate operation for ascribing uncertainty to the union of two sets. There have been various justifications given for taking max, but a discussion of this issue is beyond the scope of this book.