1.38 Consider estimating the ratio /; of two multinomial parameters. Should the estimate depend at all on...

1.38 Consider estimating the ratio πί/π; of two multinomial parameters. Should the estimate depend at all on the counts in other categories?

a. With a frequentist approach, explain why the ML estimate of πί/π; is n₁/nj.

b. For a Dirichlet prior, show that using the Bayes estimates of π; and π; to estimate

πί/π; uses also the counts in other categories. (However, the posterior distribution of y = π₁/(πί + π;) is the same as its posterior distribution ignoring the other counts and treating y₁ as binomial with sample size (y + yj) and parameter y.)