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    Countably additive Kolmogorovian probability distribution... — Carmelics
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    Countably additive Kolmogorovian probability distributions must violate regularity when the sample space is uncountable

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.A Kolmogorovian probability function is real-valued
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    • 2.Real-valued probability functions cannot assign positive probability to every point in an uncountable sample space while remaining countably additive and summing to 1
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Hyperreal-valued probability functions (as developed by Bernstein & Wattenberg 1969) assign infinitesimal positive probability to each point in uncountable spaces.
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    • 2.Regularity is preserved under hyperreal probability when countable additivity is replaced by the weaker condition of hyperfinite additivity.
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    • 3.Therefore the conflict between regularity and uncountable sample spaces is an artifact of restricting probability values to the reals, not a necessary feature of probability theory.
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    Reason against 2 of 2
    ?
    • 1.Kolmogorov's axioms presuppose a sigma-algebra over the sample space, which need not include every singleton of an uncountable set.
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    • 2.Regularity is only violated when undefined singletons are illicitly treated as measurable events requiring positive probability assignment.
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    • 3.The claim conflates the measure-theoretic structure of a probability space with metaphysical claims about possibility, committing a category error identified by Williamson (2007).
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    Related

    A Kolmogorovian probability function is real-valuedHyperreal-valued probability functions (as developed by Bernstein & Wattenberg 1...Kolmogorov's axioms presuppose a sigma-algebra over the sample space, which need...Real-valued probability functions cannot assign positive probability to every po...
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    Regularity is only violated when undefined singletons are illicitly treated as m...Regularity is preserved under hyperreal probability when countable additivity is...The claim conflates the measure-theoretic structure of a probability space with ...Therefore the conflict between regularity and uncountable sample spaces is an ar...

    Similar

    Real-valued probability functions cannot assign positive probability t...81%Continuous probability distributions require a restriction to countabl...76%Regularity can be preserved for uncountable sample spaces by using hyp...73%Likelihood inference can fail to be statistically consistent71%

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    We have seen a striking violation of regularity in de Finetti’s lottery: his assignment of 0 to each ticket. Regularity may be preserved here by countably additive probabilities, but at the expense of a uniform distribution—for example, \(\frac{1}{2}\) to ticket 1, \(\frac{1}{4}\), to ticket 2, \(\frac{1}{8}\) to ticket 3, and so on. It may be shown that if \(F\) is uncountable, a Kolmogorovian (real-valued) probability distribution must violate regularity. (See e.g. Hájek 2003b.) This has led
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    Details

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    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit