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52 3. Newton’s Method and Scoringdo not appear in most population samples. In the absence of data to thecontrary, one can argue that it is reasonable to steer haplotype frequencyestimates toward linkage equilibrium. From an empirical Bayesian perspective,the most natural equilibrium frequencies can be found by computingallele frequency estimates at each locus and taking products. We build onthis insight by choosing a Dirichlet prior whose mode occurs at these estimatedhaplotype frequencies. A short calculation shows that the modeof the Dirichlet density (3.12) reduces to the point p with coordinatesp i = β i /β, where β i = α i − 1 and β = α . − k. Thus we choose β i sothat the ratio β i /β coincides with the frequency of the ith haplotype underlinkage equilibrium using the estimated allele frequencies. These choices donot determine β, which specifies the overall strength of the prior.Problem 5 of Chapter 2 discusses the standard EM algorithm for maximumlikelihood estimation of haplotype frequencies from a random sampleof individuals. The Bayesian version of the EM algorithm adds β pseudohaplotypesto the various haplotype classes in proportion to their linkageequilibrium frequencies β i /β. Problem 12 of this chapter shows how to includethese pseudo-haplotypes in the haplotype counting update of the EMalgorithm.3.9 Problems1. Let f(x) be a real-valued function whose Hessian matrix (∂ 2∂x i∂x jf)is positive definite throughout some convex open set U of R m .Foru ≠ 0 and x ∈ U, show that the function t → f(x + tu) of the realvariable t is strictly convex on {t : x + tu ∈ U}. Use this fact todemonstrate that f(x) can have at most one local minimum point onany convex subset of U.2. Apply the result of Problem 1 to show that the loglikelihood of theobserved data in the ABO example of Chapter 2 is strictly concaveand therefore possesses a single global maximum. Why does the maximumoccur on the interior of the feasible region?3. Show that Newton’s method converges in one iteration to the maximumof the quadratic functionL(θ) = d + e t θ + 1 2 θt Fθif the symmetric matrix F is negative definite.4. Verify the loglikelihood, score, and expected information entries inTable 3.1 for the binomial, Poisson, and exponential families.