Suspending judgement

You are sometimes in a position where none of your evidence seems to speak for or against the truth of some proposition. Arguably, a reasonable attitude to take towards such a proposition is suspension of judgement.

اقتباس :: When there is little or no information on which to base our conclusions, we cannot expect reasoning (no matter how clever or thorough) to reveal a most probable hypothesis or a uniquely reasonable course of action. There are limits to the power of reason. (Walley 1991: 2)

Consider a coin of unknown bias. The Bayesian agent must have a precise belief about the coin’s landing heads on the next toss. Given the complete lack of information about the coin, it seems like it would be better just to suspend judgement. That is, it would be better not to have any particular precise credence. It would be better to avoid betting on the coin. But there just isn’t room in the Bayesian framework to do this. The probability function must output some number, and that number will sanction a particular set of bets as desirable.
Consider [ltr]P−(X)[/ltr] as representing the degree to which the evidence supports [ltr]X[/ltr]. Now consider [ltr]I(X)=1−(P−(X)+P−(¬X))[/ltr]. This measures the degree to which the evidence is silent on [ltr]X[/ltr]. Huber (2009) points out that precise probabilism can then be understood as making the claim that no evidence is ever silent on any proposition. That is, [ltr]I(X)=0[/ltr] for all [ltr]X[/ltr]. One can never suspend judgement. This is a nice way of seeing the strangeness of the precise probabilist’s attitude to evidence. Huber is making this point about Dempster-Shafer belief functions (seehistorical appendix, section 7), but it carries over to IP in general.
The committed precise probabilist would respond that setting [ltr]p(X)=0.5[/ltr] is suspending judgement. This is the maximally noncommittal credence in the case of a coin flip. More generally, suspending judgement should be understood in terms of maximising entropy (Jaynes 2003; Williamson 2010: 49–72). The imprecise probabilist could argue that this only seems to be the right way to be noncommittal if you are wedded to the precise probabilist representation of belief. That is, the MaxEnt approach makes sense if you are already committed to representation of belief by a single precise probability, but loses its appeal if credal sets are available. Suspending judgement is something you do when the evidence doesn’t determine your credence. But for the precise probabilist, there is no way to signal the difference between suspension of judgement and strong evidence of probability half. This is just the weight/balance argument again.
To make things more stark, consider the following delightfully odd example from Adam Elga:

اقتباس :: A stranger approaches you on the street and starts pulling out objects from a bag. The first three objects he pulls out are a regular-sized tube of toothpaste, a live jellyfish, and a travel-sized tube of toothpaste. To what degree should you believe that the next object he pulls out will be another tube of toothpaste? (2010: 1)

In this case, unlike in the coin case, it really isn’t clear what intuition says about what would be the “correct” precise probabilist suspension of judgement. What Maximum Entropy methods recommend will depend on seemingly arbitrary choices about the formal language used to model the situation. Williamson is well aware of this language relativity problem. He argues that choice of a language encodes some of our evidence.
Another response to this argument would be to take William James’ response to W.K. Clifford(Clifford 1901; James 1897). James argued that as long as your beliefs are consistent with the evidence, then you are free to believe what you like. So there is no need to ever suspend judgement. Thus, the precise probabilist’s inability to do so is no real flaw. This attitude, which is sometimes called epistemic voluntarism, is close to the sort of subjectivism espoused by Bruno de Finetti, Frank Ramsey and others.
There does seem to be a case for an alternative method of suspending judgement in order to allow you to avoid making any bets when your evidence is very incomplete, ambiguous or imprecise. If your credences serve as your standard for the acceptability of bets, they should allow for both sides of a bet to fail to be acceptable. A precise probabilist cannot do this since if a bet has (precise) expected value [ltr]e[/ltr] then taking the other side of that bet (being the bookie) has expected value [ltr]−e[/ltr]. If acceptability is understood as nonnegative expectation, then at least one side of any bet is acceptable to a precise agent. This seems unsatisfactory. Surely genuine suspension of judgement involves being unwilling to risk money on the truth of a proposition at any odds.
Inspired by the famous “Bertrand paradox”, Chandler (2014) offers a neat argument that the precise probabilist cannot jointly satisfy two desiderata relating to suspension of judgment about a variable. First desideratum: if you suspend judgement about the value of a bounded real variable [ltr]X[/ltr], then it seems that different intervals of possible values for [ltr]X[/ltr] of the same size should be treated the same by your epistemic state. Second desideratum: if [ltr]Y[/ltr] essentially describes the same quantity as [ltr]X[/ltr], then suspension of judgement about [ltr]X[/ltr] should entail suspension of judgement about [ltr]Y[/ltr]. Let’s imagine now that you have precise probabilities and that you suspend judgement about [ltr]X[/ltr]. By the first desideratum, you have a uniform distribution over values of [ltr]X[/ltr]. Now consider [ltr]Y=1/X[/ltr]. [ltr]Y[/ltr] essentially describes the same quantity that [ltr]X[/ltr] did. But a uniform distribution over [ltr]X[/ltr] entails a non-uniform distribution over [ltr]Y[/ltr]. So you do not suspend judgement over [ltr]Y[/ltr]. A real-world case of variables so related is “ice residence time in clouds” and “ice fall rate in clouds”. These are inversely related, but describe essentially the same element of a climate system (Stainforth et al. 2007: 2154).
So a precise probabilist cannot satisfy these reasonable desiderata of suspension of judgement. An imprecise probabilist can: for example, the set of all probability functions over [ltr]X[/ltr] satisfies both desiderata. There may be more informative priors that also represent suspension of judgement, but it suffices for now to point out that IP seems better able to represent suspension of judgement than precise probabilism. Section 5.5 of Walley (1991), discusses IP’s prospects as a method for dealing with suspension of judgement

Suspending judgement

Suspending judgement :: تعاليق

Suspending judgement