The fiducial argument allows construction of a probabilit... — Carmelics

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The fiducial argument allows construction of a probability distribution over parameter values based on the observed sample.

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2 reasons against

Reasons For

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Reason for

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1.The pivotal quantity \(\hat{\theta}(s) - \theta\) has a known distribution (normal with the aforementioned variance).
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2.The distribution of the pivotal quantity is independent of the sample.
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3.Fixing the sample to \(s\) fixes the value of \(\hat{\theta}\), which uniquely determines a distribution over the parameter values \(\theta\).
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Reasons Against

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Reason against 1 of 2

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1.The fiducial argument illicitly treats θ as a random variable after conditioning on data, violating the frequentist prohibition on parameter distributions.
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2.Fixing the observed sample s does not transform a sampling distribution over statistics into a legitimate probability distribution over fixed parameters without a prior.
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3.Savage, Lindley, and others demonstrated that fiducial distributions are not coherent probabilities—they fail additivity when derived from multi-dimensional pivots.
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Reason against 2 of 2

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