Suppose that data (X1,...,XK) follows a (singular) Multinomial distribution withparameter (θ1,...,θK) withK≥3. Also suppose that the parameterθ= (θ1,...,θK)has a DirichletK(α1,...,αK) distribution. Letγ=θ1/(θ1+θ2).1 (a) Write down the marginal prior distribution ofγ. [Hint:First consider themarginal prior for (θ1,θ2) and then perform a change of variable: (θ1,θ2) to(γ,λ), whereλ=θ1+θ2.](b) Derive the marginal posterior distribution ofγ. [Hint:First derive the jointposterior distribution of (θ1,...,θK), then the marginal posterior distribution of(θ1,θ2), and finally use the general fact obtained in (a).](c) Show that the posterior distribution in (b) is identical to the posterior distributionofγobtained by treatingX1as an observation from a Binomial distribution withsample size =X1+X2and success probabilityγ. Interpret the result
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