3.47 Consider a particular choice of Dirichlet means {Yij = E(j) = j/K} for the Bayes estimator...

3.47 Consider a particular choice of Dirichlet means {Yij = E(π¡j) = α¡j/K} for the Bayes estimator (1.19) extended to two-way tables. Show that the total mean squared error is 2

[K/(n + K)]² [Σ(πιj - Vij)²] + [n/(n + K)]² [1 - Σπ]

divided by n. Show that the value of K that minimizes this is 2

K = (1 -Σπ)/ [Σ(Vij - πι)2].

Fienberg and Holland (1973) showed this and used the empirical Bayes approach of estimating K by replacing n by the sample proportion p and letting {Yij = Pi+P+j}.

Albert (2010) surveyed Bayesian methods for smoothing contingency tables.