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. week 6

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. week 6

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 week 6

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….. week 6