Abstract
One sometimes sees it suggested that the phenomenon of dilation is a problem for sets of probabilities, since your belief state becomes “less informed” on learning new evidence, and that is something that should not happen. This article explores several ways to make precise the idea of “informativeness” for a set of probabilities, and finds that this criticism of imprecise probability does not stand up to scrutiny.
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Notes
- 1.
The historical appendix to my SEP entry (Bradley 2014) owes much to Teddy’s prompting.
- 2.
This starting point immediately betrays my allegiance: I am approaching this topic from the point of view of philosophy and formal epistemology, rather than from a more purely mathematical position. If I were committed to the latter, I would prefer to talk in terms of lower previsions or sets of desirable gambles, and if I did so, I would perhaps be less bothered by dilation. It really does seem that dilation, if it is a problem at all, is a problem for a credal set approach rather than other nearby formal frameworks. See Gregory Wheeler’s contribution to this volume for more discussion of the alternatives, and also Augustin et al. (2014) and Troffaes and De Cooman (2014).
- 3.
For more on characterising the exact nature of dilation, and the circumstances under which it occurs, see Herron et al. (1994), Seidenfeld and Wasserman (1993), Pedersen and Wheeler (2014), Herron et al. (1997), Pedersen and Wheeler (2015), Pedersen and Wheeler (2019), and Nielsen and Stewart (2021).
- 4.
One might think that it should be \(\left \{\frac {1}{16},\dots ,\frac {15}{16}\right \}\) instead. Whether or not we demand convexity won’t make a difference in what follows, and the interval is a slightly neater notation.
- 5.
As we’ll see, I think this way of talking—taking \(\mathbb {Q}(B)\) to be an adequate representation of your confidence in B—is problematic, but let’s roll with it for now.
- 6.
I don’t think many would balk at Good’s modest assumptions in the case of a finite partition. As Kadane et al. (2008) point out, countable additivity is required for the result to hold in the infinite case, and, in the context of the current volume, it would be remiss of me not to point out that that is not a principle that enjoys universal acceptance.
- 7.
- 8.
Although Bradley and Steele (2014) argue that the alternatives are no better.
- 9.
See Pedersen and Wheeler (2014) for a more sophisticated discussion of dilation and irrelevance.
- 10.
This is one point at which it is clear that my main target is the pro-precise probability crew, rather than, the anti-pointwise-conditioning posse.
- 11.
Or better yet—to avoid what (Easwaran 2014) calls the “numerical fallacy”—it is the whole probability space (including a representation of the algebra of events) that captures your epistemic state.
- 12.
- 13.
This, in essence, is the notion of “informativeness” from De Cooman (2005).
- 14.
- 15.
Stewart and Nielsen (this volume) also discuss this kind of approach to measuring the size of credal sets. For another approach to uncertainty roughly in this vein, see Chambers and Melkonyan (2007).
- 16.
Δ and Φ have been chosen as labels for the partitions in order to be mnemonic: Δ is the dilating partition, Φ is the finer partition.
- 17.
Here and throughout the logarithms are base 2 logarithms, and by convention \(0\log 0 = 0\).
- 18.
This essentially follows from the fact that H(L|M) < H(L) for random variables L,M.
- 19.
We could then write \(AU(\mathbb {P},\varDelta )\) as \(\overline {H}(\mathbb {P},\varDelta )\).
- 20.
The lower bound is attained at, for example \(p(B) = \frac {1}{16}\) where \(H(p,\varDelta ) = -\frac {1}{16}\log {\frac {1}{16}} - \frac {15}{16}\log {\frac {15}{16}}\). As p(B) tends to 0, H(p, Δ) tends to 0.
- 21.
Lower bound attained at, for instance \(p(BL)=\frac {1}{32}\). As p(BL) tends to zero, H(p, Φ) tends to one.
- 22.
As \( \underline {\mathbb {Q}}(B)\) tends to 0, \(CSU(\mathbb {Q},\varDelta )\) tends to infinity.
- 23.
There is a slight abuse of notation here, since p(−|L) is actually a member of \(\mathbb {Q}\) rather than \(\mathbb {Q}_L\). I do things this way to emphasise what’s going on. In any case, since conditionalisation is rigid, for all \(p\in \mathbb {Q}_L\), we have p(X|L) = p(X) for all X.
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Bradley, S. (2022). Dilation and Informativeness. In: Augustin, T., Cozman, F.G., Wheeler, G. (eds) Reflections on the Foundations of Probability and Statistics. Theory and Decision Library A:, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-031-15436-2_6
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