In Chapter2, we studied the AGM revision operators obtained via theτ-rule, which consisted in generating a system of stable sets from the prior, and then revising the doxastic state via the resulting ordering on states. There is a sense in which this procedure does not exploit much of thedynamics of probabilistic credences: revisions depend entirely on this initial ordering. By contrast, τ-generated revision operators faithfully describe the dynamics of probabilistically stable reasoning under Bayesian conditioning: the τ-rule is applied at each conditioning step, and the changes in the agent’s beliefs closely mirror the changes that conditioning brings to the structure of stable sets – such as, crucially, the formation of new stable sets (spheres).
The most evident distinction between those two revision mechanisms is of course that
τ-generated revisions, by their very design, commute with Bayesian conditioning, while the AGM procedure above does not, as we discussed at length in the previous chapter.
On the logical side, another difference between AGM and τ-generated revision is that the latter fails the (Or) rule, while our AGM-driven revision always satisfies (Or) (it is a straightforward exercise to show that this holds for any AGM operator). This failure is somewhat troubling, and some may see it as quite a damning feature ofτ-generated revision.
The usual arguments for the (Or) rule present it as a commonsense desideratum for handlingcase reasoning. The rationale for case reasoning is best illustrated in the context of arguments for the (weaker) rule (exOr). Suppose we have ϕ |∼ψ and ¬ϕ |∼ψ, while also having ϕ∨ ¬ϕ6 |∼ψ. This means that the agent believesψ conditionally on learning either ϕ
or ¬ϕ, but does not currently acceptψ. This seems to run counter to the following intuitive principle, expressed by Lin and Kelly:
if you know that you will accept a proposition regardless what you learn, you should accept it already. [25, p. 960]
The rule can also be seen as analogous to Savage’s decision-theoretic sure-thing principle [48]: if, in a decision problem, the agent knows she would select action a conditional on eventX being true, and she would select action aconditional on its negationXc, then she ought to select actiona outright.
A natural way to excuse the failure of the (Or) rule consists in marking the distinction betweenhypothetical conditionals and actual update conditionals. On the first reading, the expressionϕ |∼ψ captures a contingency plan that is transparent to the agent herself: under this reading, the agent knowingly commits to acceptingψ in both cases ϕand ¬ϕ. Refusing to acceptψ appears incongruous, since she also knows that one ofϕ and¬ϕ already obtains (even though she may not know which).
On the second reading, in the conditional ϕ |∼ψ, the antecedentϕ ought to be read as a truly dynamic operator: in a spirit closer to dynamics logics [45,5], we could conceive of learningϕas an event taking place. Following the dynamic tradition, we can suggestively write this informational event as [!ϕ] to distinguish it from ϕ, the mere propositionthatϕ
holds. Then the formulaϕ |∼ψ – properly read as [!ϕ]ψ – expresses that the informational state of the agent is such that, after actually, truthfully learning ϕ, she acceptsψ. It may be the case that the information events [!ϕ] and [!¬ϕ] both lead her to acceptψ; but, at the current stage, neither event actually happened, so nothing prompted the agent to adopt that belief. On this reading, there is no rationality failure on the part of the agent: for one thing, she need not believe that either event [!ϕ] or [!¬ϕ] will occur (even though she knows that eitherϕor¬ϕis the case). For another, the outcomes of learning events need not be transparent to her.
In what follows, we will remain neutral on this interpretative issue. In particular, we will not pursue in detail this second reading here: it is most fruitful to first understand the be- haviour of basic stability-induced conditionals, regardless of matters of interpretation, before we can resort to more expressive logics that capture the distinction between propositions
ϕand their corresponding learning events [!ϕ]. In any case, while the failure of (Or) may be unsettling, it also renders the logic of probabilistic stability an unusual and interesting object of study.
On the other hand, probabilistically stable revision can outperform AGM revision in probabilistic learning tasks. This important difference between AGM operators and probabilistically stable revision emerges when considering simple statistical learning scenarios. Consider the following:
Example 3.2.3
There are two urns: one urn with only white balls (call it W), and one with black and white balls in equal proportions (call itBW). The learner is presented with urn W, but she does not know which of the two urns she was given: her priors assign probability 1/2 to each urn, and assumes the draws are independent and identically distributed (with the obvious Bernoulli distribution corresponding to each urn: the urn Wcorresponds to the parameter
p= 1 for the probability of a white ball, and BWcorresponds to the parameter p= 1/2). She repeatedly draws balls from the urn, with replacement, and adjusts her credences accordingly via conditioning. In this context, the learner can select a belief revision procedure to be followed in parallel to the updating of her subjective probability measure. Here we compare two such procedures.
The first procedure is the one discussed in Chapter 2, which consists in generating a system of spheres form the prior, and then revising her doxastic state via the resulting AGM revision operation. The second one is τ-generated revision: at each draw of a ball from the urn, the agent conditions on the current evidence, after which she applies the τ-rule to determine her doxastic state. Now, the question is: which of those two procedures will allow the agent to learn the correct urn? That is, as the agent draws white ball after white ball, is there a time at which she will come to believe that the urn is W, and stick to this belief forever after?
We represent each possible sequence of draws as a binary sequence, with a draw of a white ball denoted by 0 and a black ball by 1. We assume that each experiment involves a finite number k of draws, and we consider what happens when we increase the number of draws in the experiment26. At each stagek, our space contains basic propositions of the form (U, x), withU∈ {W,BW} and x∈ {0,1}k, representing the draw of sequence x from urnU. Thus, the hypothesis W is identified with {(W, x)|x∈ {0,1}k}.
Now, consider the AGM reasoner. Note that, given her prior distribution µ, there exist no stable sets X of measure less that 1. This is because, in order to exceed the threshold
t, any stable set S would have to contain at least one state of the form (BW, x), for some
x∈ {0,1}k, where µ(BW, x) =µ(BW)·µ(x|BW) =1 2 · 1 2k = 1 2k+1. 26
This is a cumbersome way of modeling the learning process here. Of course, it is much more natural to model this example using a single infinite space, e.g. by taking a sample space{W,BW} ×2ω and an algebra generated by basic opens of the form (S,[[x]]) withS∈ {W,BW}and [[x]] ={X∈2ω|X extendsx}. However, in the infinite case there are some subtleties to address concerning the fact the space is nonatomic, which trivialises the notion of stability (as any set with measure below one has defeaters – see brief discussion in Chapter4). There is also no least set of measure 1, and so we need to modify the notion of acceptance slightly. In order to avoid those difficulties, it is more convenient here to model the scenario as involving increasingly large but finite spaces representing increasing numbers of draws.
Since µ(S)<1, there must also be a state of the form (BW, y) in Sc, where we also have
µ(BW, y) = 1/2k+1. But then
µ(S|Sc∪ {(BW, x)})≤ µ(BW, x)
µ(BW, x) +µ(BW, y) = 1/2,
and so the proposition Sc∪ {(BW, x)} is a defeater forS.
Thus, the AGM-complying reasoner starts by believing only the strongest probability 1 proposition, which is consistent with(BW,0. . .0)(that is, it is consistent with the proposition that only white balls are drawn from the mixed urn BW). But then theBW hypothesis will never be eliminated by AGM revision, since any piece of information received by further draws – namely, any sequence of 0’s – remains logically consistent with BW. Thus, the mixed-urn hypothesis is never set aside and so W cannot be learnt.
Using the τ-generated revision, on the other hand, allows the reasoner to learn that the urn is W. After n trials, the learner observes a sequence 0(n):= (0, . . . ,0) ofn white balls. Her credence in W is then
µ(W|0(n)) = 1/2
1/2 + 1/2n+1 =
1 1 + 1/2n
and thus, limn→∞µ(W|0(n)) = 1. At some point, we will pass the stability threshold t
and have µ(W,0(k))>1−tt·µ(BW,0(k)): the agent comes to believe W, her strongest stable proposition being the singleton set{(W,0(k))}.
As this example illustrates, there exist cases where AGM-driven revision never learns the correct outcome, while stability reasoning does.