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"mun I *; 5:1 3. Suppose that data (X1, .. . ,XK) follows a (singular) Multinomjal distribution with parameter (:91. . . .,6'K) with K 2 3. Also...

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Q3

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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“mun I *; 5:1 3. Suppose that data (X1, .. . ,XK) follows a (singular) Multinomjal distribution with
parameter (:91. . . .,6'K) with K 2 3. Also suppose that the parameter 3 21:311.”,ng
has a Dirichletgwl, . .. ,oghr] distribution Let 1-“ = HJflBI + 32). 1 (a) Write down the marginal prior distribution of '7. [l-Iint: First consider the
marginal prior for (61,32) and then perform a change of variable: (91,32) to
(*7, A]? where A = 91 + 32.] (b) Derive the marginal posterior distribution of '7. [Hint: First derive the joint
posterior distribution of {B1, . .. , 93'), then the marginal posterior distribution of
{31? 62), and finally use the general fact. obtained in (a]] (c) Show that the posterior distribution in (b) is identical to the posterior distribution
of 7 obtained by treating X1 as an observation from a Binomial distribution with
sample size = X; + X2 and success probability '7. Interpret the result. 4. Modified from Exercise 3.3 of BDAB: On September 25, 1988, the evening of a
presidential campaign debate, ABC news conducted a survey of registered voters in
the United States; 639 persons were polled before the debate? and a diEerent group of
639 persons were polled after. The results are displayed in the following table. Survey Bush Dukakis No opinion f other Total
pro-debate 294 307 38 639
postdebate 238 332 19 639

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