Implement the Metropolis algorithm. Parameter for binomial distribution is probability

of success theta [0, 1], n = 20. Assume the observed data vector gives Sn = 5.

(a) (5 points) Assume the prior distribution as in our lecture, pi(theta) = 2 cos^2(4pi*theta). Gener-

ate samples from the posterior distribution pi(theta|Y ). Discretize thetato be a uniform grid

of points [0, 1/10, . . . , 1]. Run the chain for n = 100, 500, 1000, and 5000 time steps,

respectively. For each time step, compare the empirical distributions with the desired

posterior distribution pi(thata|Y ). (Hint: you may use ergodicity: hence the distribution

of states can be estimated from one sample path when the number of time steps is

large (e.g. 500).)

(b) (10 points) Following from the previous question, evaluate the mean of the poste-

rior distribution (this gives an estimator for the parameter value), and Epi(thata|Y )f[

thata-1/2]^2 =

sum

(theta - 1/2)^pi(theta|Y )d theta.

(c) (10 points) Now assume the the prior distribution is given by pi(theta) is a uniform

distribution over [0, 1]. Repeat the above questions, (a) and (b).