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It is not the case that The Likelihood Ratio Convergence Theorem presupposes a fixed, well-defined hypothesis space that persists through evidential accumulation.
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Reasons For
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Reason for
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1.
Real inquiry involves discovering new hypothesis classes (paradigm shifts, novel frameworks) that weren't in the original space.
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2.
The theorem assumes idealized conditions rarely met in practice; actual evidence often suggests the hypothesis space itself was incomplete or misconstrued.
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3.
Requiring fixed hypothesis spaces privileges whatever initial space was proposed, potentially excluding better-framed alternatives discovered later.
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Reasons Against
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Reason against
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1.
The theorem's mathematical proof requires hypothesis space to remain constant; changing it mid-analysis invalidates the convergence guarantee.
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2.
Scientific practice often treats hypothesis spaces as fixed within experimental design to ensure rational belief updating.
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3.
Without a stable space, we cannot distinguish between evidence against a hypothesis and evidence that requires reformulating hypotheses entirely.
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