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    Continuous probability distributions require a restrictio... — Carmelics
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    Home/Modality & Possibility
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    Continuous probability distributions require a restriction to countable additivity rather than full additivity

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.In continuous distributions there are uncountably many states
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    • 2.Each individual state in a continuous distribution has probability 0
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    • 3.Events containing uncountably many states often have non-zero probability
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.de Finetti's coherence framework demonstrates that full additivity can be preserved by treating probability as a finitely additive measure without requiring countable additivity at all.
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    • 2.Restricting to countable additivity is itself a conventional stipulation, not a mathematical necessity, as Dubins and Savage showed in 'How to Gamble If You Must' (1965).
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    • 3.The alleged violation of full additivity presupposes that uncountable sums must behave like countable ones, but this conflates two distinct mathematical regimes with different convergence properties.
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    Reason against 2 of 2
    ?
    • 1.Nonstandard analysis, developed by Abraham Robinson, assigns infinitesimal but non-zero probabilities to individual points in continuous distributions, dissolving the tension without restricting additivity.
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    • 2.If each point receives a positive infinitesimal probability and hyperfinite summation is employed, full additivity is recoverable within a well-defined mathematical framework endorsed by Bernstein and Wattenberg (1969).
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    Modality & Possibility

    Related

    Assigning probability 0 to each individual state while allowing uncountably many...Each individual state in a continuous distribution has probability 0Events containing uncountably many states often have non-zero probabilityIf each point receives a positive infinitesimal probability and hyperfinite summ...
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    In continuous distributions there are uncountably many statesNonstandard analysis, developed by Abraham Robinson, assigns infinitesimal but n...Restricting to countable additivity is itself a conventional stipulation, not a ...The alleged violation of full additivity presupposes that uncountable sums must ...de Finetti's coherence framework demonstrates that full additivity can be preser...

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    However, many applications of probability require what are known as continuous distributions (such as the uniform/rectangular, normal, and beta distributions), and thus require a restriction to countable additivity. In a continuous distribution, there are uncountably many states, usually named by real numbers. Each individual state has probability 0, even though events containing uncountably many states often have non-zero probability. (This violates full additivity.) However, in the common cont
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    Details

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