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Bayesian Statistics: An Introduction, 4th Edition by Peter M. Lee

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2.12 Normal mean and variance both unknown

2.12.1 Formulation of the problem

It is much more realistic to suppose that both parameters of a normal distribution are unknown rather than just one. So we consider the case where we have a set of observations  which are  with θ and both unknown. Clearly,

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from which it follows that the density is in the two-parameter exponential family as defined above. Further

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wherein we define

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(rather than as  as in the case where the mean is known to be equal to μ). It is also convenient to define

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(rather than s2=S/n as in the case where the mean is known).

It is worth noting that the two-dimensional vector or equivalently is clearly sufficient for given ...

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