21. To further study the relationship between identifiability and MCMC convergence, consider again the two-parameter likelihood model Y N(1 + 2 , 1) ,

21. To further study the relationship between identifiability and MCMC convergence, consider again the two-parameter likelihood model Y ∼ N(θ1 + θ2 , 1) , with prior distributions θ1 ∼ N(a1, b2 1) and θ2 ∼ N(a2, b2 2), θ1 and θ2 independent.

(a) Clearly θ1 and θ2 are individually identified only by the prior; the likelihood provides information only on μ = θ1 + θ2. Still, the full conditional distributions p(θ1|θ2, y) and p(θ2|θ1, y) are available as normal distributions, thus defining a Gibbs sampler for this problem.

Find these two distributions.

(b) In this simple problem, we can also obtain the marginal posterior distributions p(θ1|y) and p(θ2|y) in closed form. Find these two distributions. Do the data update the prior distributions for these parameters?

(c) Set a1 = a2 = 50, b1 = b2 = 1000, and suppose we observe y = 0.

Run the Gibbs sampler defined in part

(a) in R for t = 100 iterations, starting each of your sampling chains near the prior mean (say, between 40 and 60), and monitoring the progress of θ1, θ2, and μ. Does this algorithm “converge” in any sense? Estimate the posterior mean of μ. Does your answer change using t = 1000 iterations?

(d) Now keep the same values for a1 and a2, but set b1 = b2 = 10. Again run 100 iterations using the same starting values as in part (b). What is the effect on convergence? Again repeat your analysis using 1000 iterations; is your estimate of E(μ|y) unchanged?

(e) Summarize your findings, and make recommendations for running and monitoring convergence of samplers running on “partially unidentified” and “nearly partially unidentified” parameter spaces.