Consider the following Bayesian model: f(, 0) X,...,n | ,0 iid N(, 0). 1/0, ...

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Consider the following Bayesian model: f(μ, 0²) ∞ X₁,...,n | μ,0² iid N(μ, 0²). 1/0², µ¤ R, o² > 0, (i) Derive an expression for the joint posterior pdf of (μ, o²) given x = (x₁,...,xn). (ii) Show that n i=1 n (x; − µ)² = Σ(x; − x)² + n(µ — ñ)² . - i=1 [2 marks] [2 marks] (iii) Using part (ii), or otherwise, show that the conditional posterior f(ulo², x) is normal. Find the mean and variance of this normal distribution. [2 marks] (iv) Show that the conditional posterior f(o²|u, x) is Inverse-Gamma. Find the shape and scale of this Inverse-Gamma distribution. [2 marks] (v) Describe how you would sample from the joint posterior f(u, o²|x). [4 marks] (vi) Describe how you construct posterior 90% credible intervals for the mean parameter μ and the variance parameter o². [2 marks] Consider the following Bayesian model: f(μ, 0²) ∞ X₁,...,n | μ,0² iid N(μ, 0²). 1/0², µ¤ R, o² > 0, (i) Derive an expression for the joint posterior pdf of (μ, o²) given x = (x₁,...,xn). (ii) Show that n i=1 n (x; − µ)² = Σ(x; − x)² + n(µ — ñ)² . - i=1 [2 marks] [2 marks] (iii) Using part (ii), or otherwise, show that the conditional posterior f(ulo², x) is normal. Find the mean and variance of this normal distribution. [2 marks] (iv) Show that the conditional posterior f(o²|u, x) is Inverse-Gamma. Find the shape and scale of this Inverse-Gamma distribution. [2 marks] (v) Describe how you would sample from the joint posterior f(u, o²|x). [4 marks] (vi) Describe how you construct posterior 90% credible intervals for the mean parameter μ and the variance parameter o². [2 marks]