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# Suppose that for the location-scale normal model described in Example 7.1.4, we use the prior…

Suppose that for the location-scale normal model described in Example 7.1.4, we use the prior formed by the Jeffreys’ prior for the location model (just a constant) times the Jeffreys’ prior for the scale normal model. Determine the posterior distribution of (µ,σ2

Suppose that  and σ > 0 are unknown. The likelihood function is then given by

i.e., the conditional prior distribution of σ  given σ2 is normal with mean µ0 and variance  Then we specify the marginal prior distribution of

Sometimes (7.1.4) is referred to by saying that σ2 is distributed inverse Gamma. The  are selected by the statistician to reflect his prior beliefs.From this, we can deduce (see Section 7.5 for the full erivation) that the posterior distribution of

To generate a value (µ,σ 2) from the posterior, we can make use of the method of composition by first generating σ2 using (7.1.6) and then using(7.1.5) to generate µ, We will discuss this further in Section 7.3.Notice that  as the prior on µ becomes increasingly diffuse,the conditional posterior distribution of µ given σ2 converges in distribution to an

Actually, it does not really seem to make sense to let  the prior distribution of the prior does not converge to a proper probability distribution. The idea here, however, is that we think of taking small,

so that the posterior inferences are approximately those obtained from the limiting posterior. There is still a need to choose α0 however, even in the diffuse case, as thelimiting inferences are dependent on this quantity