A class of sampling distribution is a conjugate family of a prior distribution, if the posterior distribution belongs to the same family for all priors and all samples. Why is this phrase incorrect?
2026-03-27 15:16:17.1774624577
conjugate prior
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Usually one says a class of priors is a conjugate family for a sampling distribution if the posterior distribution belongs to the same family for all samples. To speak of the sampling distribution and the posterior distribution as belonging to the same family, rather than of the prior and posterior distributions belonging to the same family, is not just wrong but weird. The samples belong to one space and the prior and posterior distributions are distributions on another space. For example, the parameter space for the family of Poisson distributions is the set of all positive numbers, whereas the samples are integer-valued. One may have a Gamma prior and a Gamma posterior, but the samples remain integer-valued. And "for all priors" is nonsense.