10 06 2024 E TS

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BAYESIAN STATISTICS A. Guglielmi & M. Gianella 10.06.2024 Properly justify all your answers. Use the indicator function to denote the support of a distr. . Exercise 1 We study the distribution of the eye color among people with black hair. There are M = 4 color categories: v1=black, v2=blue, v3=brown and v4=green. We rule out cases with different colors of the two eyes. We select a sample of people (with black hair) of sizen and assume that the color of the eyes of each individual is described by iid random variables, conditional to parametersp := (p1, . . . , p4), where p1+· · ·+p4 = 1 and pj > 0 for all j = 1, . . . ,4. Let Y = (Y1, . . . , Y4) the vector of the associated counts, i.e., Yj is the number of individuals in the sample of size n with eye color in the j-th category. It is well-known that Y ∼Multinomial(n, p), that is P(Y1 = y1, . . . , Y4 = y4|p) = n! y1! · · · y4! py1 1 · · · py4 4 , if 4X i=1 yi = n. (1) 1. Find a conjugate prior density π for p under model (1). Denote by α = (α1, . . . , α4) its hyperparameters, writing down the set of values they can assume. Now assume n = 50, α1 + α2 + α3 + α4 = 5 and α2 = α3 = 1. 2. Fix the hyperparameter α1 of the prior π found at point 1. such that the marginal prior variance of Y1 is equal to 20. 3. We have registered the eye color of people with black hair, obtaining y1 = 28, y2 = 10, y3 = 7. Find the posterior mean of each pj, j = 1, 2, 3, 4, using available data and the prior derived above. Compute also the posterior variance of p1 and of p2. 4. Test the hypotheses that the percentage of blue eyes among black hair people is larger than 20% versus the percentage of blue eyes among black hair people is smaller or equal than 20%. Compute the Bayes factor and draw your conclusion. (Hint: use a proper approximation…