Question

QUESTION B

5. (a) Describe the Dirichlet-Multinomial model. In particular, define the Likelihood, the prior and posterior distributions. Additionally, suggest a non-informative prior. 5 marks) ) A sample of n 1000 women were interviewed about their preferences on three brand of eyeliners. The outcome is the following: 1st Brand234 preferences 338 preferences 2nd Brand 428 preferences 3rd Brand Suppose that this dataset is analyzed with a Dirichlet-Multinomial model. Estimate the posterior probabilities of selecting a certain brand when a Dir(40,20,30) prior is assumed. [5 marks

Answer 1

We now state the general form of the multinomial formula. Suppose that an experiment has r possible outcomes, so the sample space is given by Ss1,S2,...,Srt Also suppose that P(si) = Pi for i = 1, 2, ,r. Then for n = ni +n2+ +nr independent trials of this experiment, the probability that each si appears ni times is given by и, P1 P2...Pr P1 P2Pr n1,n2 , . . . ,nr

So in this case we have to use the above formulae

P(D(40,20,30))= (40+20+30)!/(40!*20!*30!) * (40/90)^40 * (20/90)^20 * (30/90)^30

=0.009667571

Console Terminal a 20+30)/(factorial (40) factorial (20) factori > factorial (40+20+30) / (factorial (40) *factorial (20) *factorial (30)) (40/90)/40 (20/90%20 (30/9OM30 * * * [1] o.009667571

Above calculation done in R.

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