Problem 3

Problem 3. Consider the following unnormalized posterior: P(91 Y) °C {01e3H-0?-03-2002-401-402} where 0 E R2. Plot the two-dimensional image of this distribution for 0 E [-5, 1012. (a) Generate an MCMC sample of size 10,000 using Metropolis algorithm. Choose the scale (A) of the proposal distribution Ar(Oi_i, A/2), where /2 is the 2 x 2 identity matrix, so that the acceptance rate is around 0.40. (i) Report the chosen scale and the actual acceptance rate; (ii) Check the convergence of the MCMC chain; (iii) Plot the samples on the image of the distribution; and (iv) Obtain the 95% equal-tailed credible intervals of the two parameters. Hint. For (ii), you may consider checking the change of the sample quantiles when more samples are generated. (b) Derive the full conditional distributions and generate 10,000 samples using Gibbs sampling. (i) Plot the samples on the image of the distribution; and (ii) Plot the marginal densities of the two parameters.