17. Consider three approaches for sampling from a N(0, 1) target density, p(x) ∝ exp(−1 2x2):

a standard Metropolis algorithm using a Gaussian proposal density, x∗|x(t−1) ∼ N(x(t−1), σ2), the simple slice sampler (3.20), and a Langevin-Hastings algorithm (3.18).

(a) Evaluate and compare the Metropolis and Langevin-Hastings acceptance ratios, the latter of which is given by (3.19).

(b) Write an R program to run single chains of 1000 iterations for each of these samplers, starting both chains at x(0) = 0. For which values of the Metropolis proposal variance σ2 do the various samplers produce the smallest lag 1 autocorrelations (i.e., faster convergence)?