A consistent set of infinite-order probabilities

David Atkinson*, Jeanne Peijnenburg

*Corresponding author for this work

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Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent.

May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of a probability of a probability of a probability, and so on, ad infinitum? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. (C) 2013 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)1351-1360
Number of pages10
JournalInternational Journal of Approximate Reasoning
Issue number9
Publication statusPublished - Nov-2013


  • Model
  • Higher-order probability
  • Infinite regress

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