Weight of evidence, balance of evidence

Evidence influences belief. Joyce (2005) suggests that there is an important difference between the weight of evidence and the balance of evidence. He argues that this is a distinction that precise probabilists struggle to deal with and that the distinction is worth representing. This idea has been hinted at by a great many thinkers including J.M. Keynes, Rudolf Carnap, C.S. Pierce and Karl Popper (see references in Joyce 2005; Gärdenfors and Sahlin 1982). Here’s Keynes’ articulation of the intuition:

اقتباس :: As the relevant evidence at our disposal increases, the magnitude of the probability of the argument may either decrease or increase, according as the new knowledge strengthens the unfavourable or the favourable evidence; but something seems to have increased in either case,—we have a more substantial basis upon which to rest our conclusion. I express this by saying than an accession of new evidence increases the weight of an argument. (Keynes 1921: 78, Keynes’ emphasis)

Consider tossing a coin known to be fair. Let’s say you have seen the outcome of a hundred tosses and roughly half have come up heads. Your degree of belief that the coin will land heads should be around a half. This is a case where there is weight of evidence behind the belief.
Now consider another case: a coin of unknown bias is to be tossed. That is, you have not seen any data on previous tosses. In the absence of any relevant information about the bias, symmetry concerns might suggest you take the chance of heads to be around a half. This opinion is different from the above one. There is no weight of evidence, but there is nothing to suggest that your attitudes to [ltr]H[/ltr] and [ltr]T[/ltr] should be different. So, on balance, you should have the same belief in both.
However, these two different cases get represented as having the same probabilistic belief, namely [ltr]p(H)=p(T)=0.5[/ltr]. In the fair coin case, this probability assignment comes from having evidence that suggests that the chance of heads is a half, and the prescription to have your credences match chances (ceteris paribus). In the unknown bias case, by contrast, one arrives at the same assignment in a different way: nothing in your evidence supports one proposition over the other so some “principle of indifference” reasoning suggests that they should be assigned the same credence (see Hájek 2011, for discussion of the principle of indifference).
If we take seriously the “ambiguity aversion” discussed earlier, when offered the choice between betting on the fair coin’s landing heads as opposed to the unknown-bias coin’s landing heads, it doesn’t seem unreasonable to prefer the former. Recall the preference for unambiguous gambles in the Ellsberg game in section 2.1. But if both coins have the same subjective probabilities attached, what rationalises this preference for betting on the fair coin? Joyce argues that there is a difference between these beliefs that is worth representing. IP does represent the difference. The first case is represented by [ltr]P(H)={0.5}[/ltr], while the second is captured by [ltr]P(H)=[0,1][/ltr].
Scott Sturgeon puts this point nicely when he says:

اقتباس :: [E]vidence and attitude aptly based on it must match in character. When evidence is essentially sharp, it warrants sharp or exact attitude; when evidence is essentially fuzzy—as it is most of the time—it warrants at best a fuzzy attitude. In a phrase: evidential precision begets attitudinal precision; and evidential imprecision begets attitudinal imprecision.(Sturgeon 2008: 159 Sturgeon’s emphasis)

Wheeler (2014) criticises Sturgeon on this “character matching” thesis. However, an argument for IP based on the nature of evidence only requires that the character of the evidence sometimes allows (or mandates?) imprecise belief and not that the characters must always match. In opposition, Schoenfield (2012) argues that evidence always supports precise credence, but that for reasons of limited computational capacity, real agents needn’t be required to have precise credences. However, her argument only really supports the claim that sometimes indeterminacy is due to complexity of the evidence and computational complexity. She doesn’t have an argument against the claims Levi, Kaplan, Joyce and others make that there are evidential situations that warrant imprecise attitudes.
Strictly speaking, what we have here is only half the story. There is a difference between the representations of belief as regards weight and balance. But that still leaves open the question of exactly what is representing the weight of evidence? What aspect of the belief reflects this difference? One might be tempted to view [ltr]P[size=13]¯¯¯¯(H)−P−(H)[/ltr] as a measure of the weight of evidence for [ltr]H[/ltr]. Walley (1991) tentatively suggests as much. However, this would get wrong cases of conflicting evidence. (Imagine two equally reliable witnesses: one tells you the coin is biased towards heads, the other says the bias is towards tails.) The question of whether and how IP does better than precise probabilism has not yet received an adequate answer. Researchers in IP have, however, made progress on distinguishing cases where your beliefs happen to have certain symmetry properties from cases where your beliefs capture evidence about symmetries in the objects of belief. This is a distinction that the standard precise model of belief fails to capture(de Cooman and Miranda 2007).[/size]
The precise probabilist can respond to the weight/balance distinction argument by pointing to the property of resiliency (Skyrms 2011) or stability (Leitgeb 2014). The idea is that probabilities determined by the weight of evidence change less in response to new evidence than do probabilities determined by balance of evidence alone. That is, if you’ve seen a hundred tosses of the coin, seeing it land heads doesn’t affect your belief much, while if you’ve not seen any tosses of the coin, seeing it land heads has a bigger effect on your beliefs. Thus, the distinction is represented in the precise probabilistic framework in the conditional probabilities. The distinction, though, is one that cannot rationalise the preference for betting on the fair coin. One could develop a resiliency-weighted expected value and claim that this is what you should maximise, but this would be as much of a departure from orthodox probabilism as IP is. If someone were to develop such a theory, then its merits could be weighed against the merits of IP type models.
Another potential precise response would be to suggest that there is weight of evidence for [ltr]H[/ltr] if many propositions that are evidence for [ltr]H[/ltr] are fully believed, or if there is a chance proposition (about [ltr]H[/ltr]) that is near to fully believed. This is in contrast to cases of mere balance where few propositions that are evidence for [ltr]H[/ltr] are fully believed, or where probability is spread out over a number of chance hypotheses. The same comments made above about resiliency apply here: such distinctions can be made, but this doesn’t get us to a theory that can rationalise ambiguity aversion.
The phenomenon of dilation (section 3.1) suggests that the kind of argument put forward in this section needs more care and further elaboration.

Weight of evidence, balance of evidence

Weight of evidence, balance of evidence :: تعاليق

Weight of evidence, balance of evidence