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On Imprecise Bayesianism in the Face of an Increasingly Larger Outcome Space

A Reply to John E. Wilcox


Wilcox proposed an argument against imprecise probabilities and for the principle of indifference based on a thought experiment where he argues that it is very intuitive to feel that one’s confidence in drawing a ball of a given colour out of an unknown urn should decrease while the number of potential colours in the urn increases. In my response to him, I argue that one’s intuitions may be unreliable because it is very hard to truly feel completely ignorant in such a situation. I further argue that Wilcox must also account for the conflicting intuition that it is absurd to have to feel completely convinced that a specific claim about reality is true in the absence of any evidence in order to avoid being irrational. It is dubious that this intuition is considerably less universal and strongly-held than Wilcox’s own intuition. Finally, I point out that even if Wilcox’s intuition were to be universally shared among members of our biological species, it is far from being clear that someone refusing to let that intuition dictate his or her beliefs would be irrational. For all these reasons, I believe that Wilcox was not successful in proving that philosophers and scientists representing uncertainty through imprecise probabilities are violating the principles of rationality.

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Fig. 1


  1. While there is still a considerable debate about the nature of physical chances in the literature, I shall just take their existence for granted here.

  2. An objective Bayesian might reply that our knowledge that there are only two possibilities is itself a piece of evidence that favours p(heads) = 0.5 over other possible values. This would, however, beg the question, as it assumes the validity of some form of the principle of indifference and presupposes that merely knowing the outcome space is as good as having tossed the coin a very large number of times. This still looks like what Salmon called “epistemological magic”.

  3. Of course, such a radical revision of her epistemology should normally take at least several weeks.

  4. Someone might object that physical chances and physical probability distributions do not really exist but are only a useful approximation that allows us to successfully model the world we live in. Even in that case, I would still believe that there is a difference between a known and unknown coin but developing that idea would require much further thought.


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I am most thankful to Dr. Seamus Bradley, Dr. Yann Benetreau-Dupin and Professor Mike Huemer for our deep and useful exchanges about the nature of subjective probabilities. I am most thankful for the insightful remarks of the anonymous reviewer regarding my article.

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Fischer, M. On Imprecise Bayesianism in the Face of an Increasingly Larger Outcome Space. J Gen Philos Sci (2022).

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  • Bayesianism
  • Principle of indifference
  • Imprecise probability
  • Weight of the argument