Question

Suppose that x = {x₁, . . . , x_m} is an independant and identically distributed sample from N(θ₁, σ²). y = {y₁, . . . , y_n} is an independant and identically distributed sample from N(θ₂, σ²). x and y are independent. The unknown parameters are θ₁∈ (−∞, ∞), θ₂∈ (−∞, ∞), and σ²∈ (0, ∞).

a) Under the improper prior p(θ₁, θ₂, σ²) ∝ (σ²)⁻², find the joint posterior distribution of (θ₁, θ₂, σ²) by finding the following three distributions:

1. p(θ₁|σ², x, y),    

2. p(θ₂|σ², x, y),    

3. p(σ²|x, y),

and show that p(θ₁, θ₂, σ²|x, y) ∝ p(θ₁|σ², x, y) × p(θ₂|σ², x, y) × p(σ²|x, y).

b) Following Part (a), let δ = θ₁− θ₂. Find a suitable change of variable for δ, such that the marginal posterior of the transformed variable from δ has a Student’s t distribution.

Answer 1