Conditional probability: suppose that if θ = 1, then y has a normal distribution with mean 1 and standard deviation σ, and if θ = 2, then y has a normal distribution with mean 2 and standard deviation σ. Also, suppose P r(θ = 1) = 0.5 and P r(θ = 2) = 0.5.
(a) For σ = 2, write the formula for the marginal probability density for y.
(b) What is P r(θ = 1|y = 1), again supposing σ = 2?
(c) Describe how the posterior density of θ changes in shape as σ is increased and as it is decreased.