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When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’…

When When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... Determine Jeffreys’ prior for the location-scale normal model and compare this with the prior used in Problem 6.5.15

Problem 6.5.15

Show that a distribution in the family When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... is completely determined once we specify two quantiles of the distribution.

Problem 6.5.17

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 When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... and σ > 0 are unknown. The likelihood function is then given by

When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

i.e., the conditional prior distribution of σ  given σ2 is normal with mean µ0 and variance When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... Then we specify the marginal prior distribution of When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

Sometimes (7.1.4) is referred to by saying that σ2 is distributed inverse Gamma. The When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... 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

When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

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 When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... as the prior on µ becomes increasingly diffuse,the conditional posterior distribution of µ given σ2 converges in distribution to an

When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’...

Actually, it does not really seem to make sense to let When is defined in Problem 6.5.15. The Jeffreys’ prior is then defined as Determine Jeffreys’... 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 the limiting inferences are dependent on this quantity