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تاريخ التسجيل : 26/10/2009
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 | |  Unknown correlations | |
Haenni et al. (2011) motivate imprecise probabilities by showing how they can arise from precise probability judgements. That is, if you have a precise probability for [ltr]X[/ltr] and a precise probability for [ltr]Y[/ltr], then you can put bounds on [ltr]p(X∩Y)[/ltr] and [ltr]p(X∪Y)[/ltr], even if you don’t know how [ltr]X[/ltr] and [ltr]Y[/ltr] are related. These bounds give you intervals of possible probability values for the compound events.For example, you know that [ltr]p(X∩Y)[/ltr] is bounded above by [ltr]p(X)[/ltr] and by [ltr]p(Y)[/ltr] and thus by [ltr]min{p(X),p(Y)}[/ltr]. If [ltr]p(X)>0.5[/ltr] and [ltr]p(Y)>0.5[/ltr] then [ltr]X[/ltr] and [ltr]Y[/ltr] must overlap. So [ltr]p(X∩Y)[/ltr]is bounded below by [ltr]p(X)+p(Y)−1[/ltr]. But, by definition, [ltr]p(X∩Y)[/ltr] is also bounded below by [ltr]0[/ltr]. So we have the following result: if you know [ltr]p(X)[/ltr] and you know [ltr]p(Y)[/ltr], then, you know[ltr]max{0,p(X)+p(Y)−1}≤p(X∩Y)≤min{p(X),p(Y)}.[/ltr] Likewise, bounds can be put on [ltr]p(X∪Y)[/ltr]. [ltr]p(X∪Y)[/ltr] can’t be bigger than when [ltr]X[/ltr] and [ltr]Y[/ltr] are disjoint, so it is bounded above by [ltr]p(X)+p(Y)[/ltr]. It is also bounded above by [ltr]1[/ltr], and thus by the minimum of those expressions. It is also bounded below by [ltr]p(X)[/ltr] and by [ltr]p(Y)[/ltr] and thus by their maximum. Putting this together,[ltr]max{p(X),p(Y)}≤p(X∪Y)≤min{p(X)+p(Y),1}.[/ltr] These constraints are effectively what you get from de Finetti’s Fundamental Theorem of Prevision (de Finetti 1990 [1974]: 112; Schervish, Seidenfeld, and Kadane 2008). So if your evidence constrains your belief in [ltr]X[/ltr] and in [ltr]Y[/ltr], but is silent on their interaction, then you will only be able to pin down these compound events to certain intervals. Any choice of a particular probability function will go beyond the evidence in assuming some particular evidential relationship between [ltr]X[/ltr] and [ltr]Y[/ltr]. That is, [ltr]p(X)[/ltr] and [ltr]p(X∣Y)[/ltr] will differ in a way that has no grounding in your evidence. 2.6 Nonprobabilistic chancesWhat if the objective chances were not probabilities? If we endorse some kind of connection between known objective chances and belief—for example, a principle of direct inference or Lewis’ Principal Principle (Lewis 1986)—then we might have an additional reason to endorse imprecise probabilism. It seems to be a truth universally acknowledged that chances ought to be probabilities, but it is a “truth” for which very little argument has been offered. For example,Schaffer (2007) makes obeying the probability axioms one of the things required in order to play the “chance role”, but offers no argument that this should be the case. Joyce says “some have held objective chances are not probabilities. This seems unlikely, but explaining why would take us too far afield” (2009: 279, fn.17). Various other discussions of chance—for example in statistical mechanics (Loewer 2001; Frigg 2008) or “Humean chance” (Lewis 1986, 1994)—take for granted that chances should be precise and probabilistic (Dardashti et al. 2014 is an exception). Obviously things are confused by the use of the concept of chance as a way of interpreting probability theory. There is, however, a perfectly good pre-theoretic notion of chance: this is what probability theory was originally invented to reason about, after all. This pre-theoretic chance still seems like the sort of thing that we should apportion our belief to, in some sense. And there is very little argument that chances must always be probabilities. If the chances were nonprobabilistic in a particular way, one might argue that your credences ought to be nonprobabilistic in the same way. What form a chance-coordination norm should take if chances and credences were to have non-probabilistic formal structures is currently an open problem.I want to give a couple of examples of this idea. First consider some physical process that doesn’t have a limiting frequency but has a frequency that varies, always staying within some interval. This would be a process that is chancy, but fairly predictable. It might be that the best description of such a system is to just put bounds on its relative frequency. Such processes have been studied using IP models (Kumar and Fine 1985; Grize and Fine 1987; Fine 1988), and have been discussed as a potential source of imprecision in credence (Hájek and Smithson 2012). A certain kind of non-standard understanding of a quantum-mechanical event leads naturally to upper probability models (Suppes and Zanotti 1991; Hartmann and Suppes 2010). John Norton has discussed the limits of probability theory as a logic of induction, using an example which, he claims, admits no reasonable probabilistic attitude (Norton 2007, 2008a,b). One might hope that IP offers an inductive logic along the lines Norton sketches. Norton himself has expressed scepticism on this line (Norton 2007). | |
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