The input problem for Jeffrey conditioning
You can't predict the stock market by looking at tea leaves. If an episode of looking at tea leaves makes you believe that the stock market will soon collapse, then -- assuming your previous beliefs did not support the collapse hypothesis, nor the hypothesis that tea leaves predict the stock market -- your new belief is unjustified and irrational. So there are epistemic norms for how one's opinions may change through perceptual experience.
Such norms are easily accounted for in the traditional Bayesian picture where each perceptual experience is associated with an evidence proposition E on which any rational agent should condition when they have the experience. But what if perceptual experiences don't confer absolute certainty on anything? Jeffrey pointed out that if there is a partition of propositions { E_i } = E_1,...,E_n such that (1) an experience changes their probabilities to some values { p_i } = p_1,...,p_n, and (2) the experience does not affect the probabilities conditional on any member of the partition, then the new probability assigned to any proposition A is the weighted average of the old probability conditional on the members of the partition, weighted by the new probability of that partition. This rule is often called "Jeffrey conditioning" and sometimes "generalised conditioning", but unlike standard conditioning it isn't a dynamical rule at all: it is a simple consequence of the probability calculus. To get genuine epistemic norms on the dynamics of belief through perceptual experience, Jeffrey's rule must be supplemented with a story about how a given experience, perhaps together with an agent's previous belief state, may fix the partition { E_i } and values { p_i } that determine a Jeffrey update. This is the "input problem" for Jeffrey conditioning.
Some initial proposals can be quickly set aside.
Most simply, one might suggest that every experience is associated with a partition { E_i } of propositions and a family { p_i } of numbers such that whenever a rational agent has the experience, her probabilities should evolve by the corresponding instance of Jeffrey conditioning. That won't do, because the new probability of E_i should be sensitive to the previous probability.
Alternatively, one might suggest that every experience is associated with a partition { E_i } and a family { x_i } of numbers such that whenever a rational agent has the experience, her probabilities should evolve by Jeffrey conditioning in such a way that the x_i specify the Bayes factors within { E_i } (Field 1978). That won't do either, in part for the reason pointed out in Garber 1980, and arguably also for the reasons mentioned in the previous post: the new probability of E_i can depend on the agent's previous probabilities for hypotheses outside { E_i }.
Weisberg (2009) argues that the input problem can't be solved at all. He offers two arguments for this conclusion, the first of which goes as follows. Under certain conditions it should make no difference to the final probability which of two experiences, call them X and Y, occurs first. However, a theorem due to Wagner (2002) shows that the update across X and Y must then proceed by the unacceptable rule of Field 1980.
Wagner's result is this: if P is updated twice by Jeffrey conditioning, once on a partition { E_i } and once on another partition { F_i }, then under some weak conditions, the final probability is independent of the order of the updates if and only if the Bayes factors for the { E_i } and the { F_i } update are preserved under changing their order, i.e. the Bayes factors stay the same no matter which of the updates occurs first.
How does this support Field's proposal? Assume it should make no difference which of X and Y occurs first. Moreover, assume that as a partial answer to the input problem, we have associated each experience with an evidence partition, say { E_i } for X and { F_i } for Y, to plug into Jeffrey's rule. By Wagner's result, we are now committed to the claim that the Bayes factors for the X update are independent of whether the update preceeds or follows the Y update (and vice versa). So commutativity of (Jeffrey-style) learning experiences with fixed partitions entails that whenever the weak assumptions of Wagner's theorem apply, then the effect of the experiences can be characterised a la Field by the associated Bayes factors.
But this doesn't quite establish what Weisberg wants to show. For one thing, we would have to show that Field's proposal yields unacceptable consequences when the preconditions of Wagner's theorem are satisfied. Interestingly, they are not satisfied in Garber's counterexample (because here { E_i } = { F_i }). More obviously, Weisberg assumes that experiences are associated with fixed evidence partitions, but this could be given up. What if X is associated with { E_i } if it comes first, and with a different partition { G_i } if it comes after Y? Isn't one of the reason why Field's attempt fails precisely that the effect an experience should have on an agent's beliefs depends on her background beliefs? If you know that the table cloth is green but are unsure about the lighting, then a perception of the cloth may directly affect only your views about the lighting; not so if instead you know about the lighting but not the colour of the cloth.
Weisberg's second argument (offered as a "diagnosis" of the first) goes as follows. Jeffrey's rule requires that probabilities conditional on evidence propositions E_i remain unchanged through an update (see condition (2) above). Weisberg argues that this does not leave room for undercutting defeaters. For suppose an experience X supports the proposition G that the tablecloth is green; later, Y reveals that L: the lighting was tricky. Y should lower the probability of G. But if the X update used the partition { G, not-G }, and if G and L were probabilistically independent with respect to the old probability, then the independence still holds for the new probability; so learning the defeater L cannot lower the probability of G.
Again, Weisberg's argument relies on an unjustified premise. Why must the first update involve the partition { G, not-G }? Indeed, the example makes clear that this is not a suitable partition for Jeffrey's rule, since the conditional probabilities are not preserved: before the update, L supported neither G nor not-G, afterwards L strongly supports not-G. So the obvious response to this argument is that the X update can't proceed by Jeffrey conditioning on { G, not-G }. It must use a more complex partition.
This still reveals something important, for it shows that the usual examples of Jeffrey conditioning are inadequate. If you see a table cloth in dim light, your beliefs should not evolve by Jeffrey conditioning on hypotheses about the cloth's colour. The cells in your evidence partition must be much more fine-grained, including hypotheses about the lighting, the functioning of your eyes, whether you are dreaming, and so on. This considerably weakens one of the main arguments in favour of Jeffrey conditioning: that it is hard to find plausible "observation sentences" that could serve as the object of strict conditioning (in particular given how informative such a sentence would have to be). The above observation shows that in this respect, Jeffrey's proposal is not much better. It also raises the worry that if the agent's probability space is not very rich, then it may be impossible to find any interesting partition that satisfies condition (2) of Jeffrey's rule.
Can we push these considerations further? What would a full solution to the input problem look like? It would specify a function f that maps any possible experience together with a prior probability distribution to an evidence partition { E_i } and a family of numbers { p_i } such that the new probability results from the prior probability by the corresponding instance of Jeffrey conditioning. This requires that
(*) the new probabilities are determined by the old probabilities together with the perceptual experience.
But there are reasons to think that (*) is false, at least within the constraints of Jeffrey's radical probabilism, where experiences don't confer certainty on "observation sentences" that distinguish them from one another.
Let me set aside some boring reasons why (*) might be false. Arguably rational beliefs are subject to other forces than perceptual input. For example they should change through practical deliberation. But this is irrelevant if we want to spell out the distinctive effect of percetual experience on rational belief. (*) may also be false because epistemic norms on how to change one's mind through perception are tolerant so that there is more than one legitimate posterior probability. If so, the "determined" in (*) should be read as "determined, to the extent that they are determined at all".
Now for the interesting reasons why I think (*) is false. Consider again the table cloth example, in Garber's version. Suppose at time t1 you catch a glimpse of the table cloth in the dimly lit room, and as a result your credence in the hypothesis that the cloth is green increases from 0.2 to 0.4. Immediately afterwards, at t2, you have an identical experience of the table cloth in the dimly lit room. Intuitively, this second experience should not significantly alter your beliefs about the cloth's colour, unlike the first experience. But can we read this off from your prior beliefs at t2, as required by (*)? Clearly the mere fact that you assign credence 0.4 to the green cloth hypothesis doesn't reveal that you previously had the very same experience: your 0.4 credence could be based on a very different experience, or on no experience at all.
Perhaps you remember the experience at t1, so that your credence function at t2 assigns positive probability to propositions constraining your experience at t1. But remember that we don't have "observation sentences". So you do not know for certain that you had an experience with special features X that tell it apart from all experiences with different cognitive effects. Nor do you even give high credence to any such hypothesis. At most we can assume that you assign high credence to a hypothesis that imperfectly characterises your experience, perhaps as an experience in which the cloth "looked somewhat more green than blue". But the fact the you assign high credence to this hypothesis does not entail enough about your previous experience to determine that your new experience at t2 should leave your beliefs essentially unchanged.
Perhaps you know that your present experience is qualitatively identical to your previous experience, although you can't spell out this qualitative character. But, first of all, how do you know this? If (*) is to be true, whatever you believe after the experience at t2 must somehow be fixed by your prior t2 beliefs and the experience at t2; we just saw that your prior beliefs do not reveal the precise qualitative character of your t1 experience; plausibly, neither does your experience at t2 -- if it does, this is at best a psychological coincidence. So if (*) is true, you can't have become confident that your new experience is qualitatively just like your previous experience. Secondly, we needn't assume that the experiences are completely indistinguishable. The problem also arises if the experiences are somewhat different, say because a cat has emerged from under the table. Still, your t2 experience should hardly affect your beliefs about the colour of the table cloth.
So one argument against (*) is that the effect of a perceptual experience on your belief state should depend on your earlier experiences, yet the relevant facts about earlier experience cannot be recovered from your prior belief state together with the new experience.
Here is another reason to think that (*) is false. Consider the time when you first see the table cloth in the dimly lit room. Given your background beliefs, for example that the lighting in the room is white, your experience should increase your degree of belief in the hypothesis that the cloth is green. By contrast, if you had thought the light is not white, your degree of belief in the green cloth hypothesis should not have increased, perhaps even decreased. But why are your beliefs about the lighting relevant? Intuitively, they are relevant because the white light hypothesis together with the green cloth hypothesis predicts that you have an experience of the kind you actually have, while the non-white light hypothesis together with the green cloth hypothesis does not make this prediction. But how can we cash this out within the constraints of Jeffrey's radical probabilism? We can't say that you previously assigned high conditional probability to getting an experience with such-and-such phenomenal character conditional on the cloth being green and the light white, for you are not supposed to have highly specific information about the phenomenal character of your experience. We could go externalist and see whether in fact, you would have the kind of experience you do have if the table cloth were green and the light white. But that would still violate (*), and it is anyway misguided because it makes your belief update depend too much on actual facts about which you may know nothing.
These arguments are not conclusive. But I think they make it quite plausible that (*) is false. So there really is no solution to the input problem for Jeffrey conditioning -- not within the constraints we have assumed. If your beliefs only pertain to ordinary propositions about the world, we cannot say how a given belief state should be affected by a given perceptual experience, because the required change is not determined by these two things.