1. Location-Scale Normal Model
Suppose that is a sample from an N(μ, σ2)
distribution where and σ > 0 are unknown. The likelihood
function is given by
Suppose we put the prior following prior on (μ, σ2). We first specify that and then specify the marginal prior distribution for σ2 as
The latter is referred to by saying that σ2 is distributed inverse Gamma.
The values of are selected by the statistician to reflect
their prior beliefs.
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2. The Posterior Distribution
The posterior distribution of (μ, σ2) is given by
and
where
To generate a value (μ, σ2) from the posterior, we begin by first generation a value for σ2 using its posterior distribution (which does not depend on μ) and then using the conditional posterior distribution of μ given σ2 to generate μ.
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3. Notes
As τ0  ∞, that is, as the prior on μ becomes increasingly diffuse,
the conditional posterior distribution of μ given σ2 converges in
distribution to the distribution because
and
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4. Estimation Suppose we are interested in . We can then use the
marginal posterior of σ2 as given above for inference, e.g., posterior
mode, posterior mean, HPD etc.
If we want to make inference about , then we need to use
its marginal prior distribution and things are not so simple. Note,
that above we could only prescribe the conditional posterior
distribution of μ given σ2.
We can approach this problem as follows…..
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