Example 6.21 (EM for the Genetic Linkage Experiment). In a genetic linkage experiment, n animals are randomly assigned (by nature) to four categories according to the multinomial distribution with pdf
f(x1, x2, x3, x4; θ) = (n! / (x1! * x2! * x3! * x4!)) * θ1^x1 * θ2^x2 * θ3^x3 * θ4^x4
where n = x1 + x2 + x3 + x4 and the cell probabilities are θ1 = 1/2, θ2 = 0/4, θ3 = (1 - θ) / 4, and θ4 = 0/4. Suppose the observed data are given as x = (1, 12, 3, 4) (125, 18, 20, 34), and we wish to obtain the MLE for θ. It is easy to check that the log-likelihood function is given by
L(θ; x) = x1 * ln(θ1) + x2 * ln(θ2) + x3 * ln(θ3) + x4 * ln(θ4) + const
Consider the genetic linkage model in Example 6.21. Show that the score and Hessian functions for θ are given by
S(θ) = (125 / θ1) + 2
38
34 and H(θ) = 125 / (θ1^2) + 2
S(θ)
Implement the Newton-Raphson procedure to find the MLE of θ. Implement a simple grid search to find the MLE. Do the Newton-Raphson, grid search, and EM approaches give the same estimate?