Forthemultinomialdistribution(2.14)withcounts {yj} in c categories satisfying j yj = n, Bayesianmethodsoftenusethe Dirichlet distribution as apriordistributionfor (1, ...,c},

Forthemultinomialdistribution(2.14)withcounts {yj} in c categories satisfying Σj yj = n, Bayesianmethodsoftenusethe Dirichlet distribution as apriordistributionfor (π1, ...,πc}, p(π1, ...,πc;α1, ...,αc) ∝ πα1−1 1 πα2−1 2 ⋯παc−1 c , 0 ≤ πj ≤ 1, Σ

j

πj = 1, for hyperparameters {αj > 0}. Thisdistributionhas E(πj) = αj~(Σk αk).

(a) Whichvaluesfor {αj} yield auniformdistributionovertheprobabilitysimplex?

(b) ShowthattheposteriordistributionisalsotheDirichlet,sotheDirichletistheconjugate prior.

(c) Showthattheposteriormeanof πj is (yj + αj)~(n + Σk αk). Withtheuniformprior, explain whytheestimatesaresampleproportionsafteraddingasingleobservationto eachcategory.