Posterior distribution as a compromise between prior information and data: let y be the number of heads in n spins of a coin, whose probability of heads is θ.

(a) If your prior distribution for θ is uniform on the range [0, 1], derive your prior predictive distribution for y,

for each k=0, 1,…, n.

(b) Suppose you assign a Beta(α, β) prior distribution for θ, and then you observe y heads out of n spins. Show algebraically that your posterior mean of θ always lies between your prior mean, and the observed relative frequency of heads

(c) Show that, if the prior distribution on θ is uniform, the posterior variance of θ is always less than the prior variance.

(d) Give an example of a Beta(α, β) prior distribution and data y, n, in which the posterior variance of θ is higher than the prior variance.