(Solution) - Let be a parameter with parameter space equal -(2025 Original AI-Free Solution)

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Let ? be a parameter with parameter space ? equal to an interval of real numbers (possibly unbounded). Let X have p.d.f. or p.f. fn(x|?) conditional on ?. Let T = r(X) be a statistic. Assume that T is sufficient. Prove that, for every possible prior p.d.f. for ?, the posterior p.d.f. of ? given X = x depends on x only through r(x).