13. Returning again to the flour beetle mortality data and model of Example 3.7, note that the decision to use Σ=2 . Σ in equation (3.17)

was rather arbitrary. That is, univariate Metropolis “folklore” suggests a proposal density having variance roughly twice that of the true target should perform well, creating bigger jumps around the parameter space while still mimicking the target’s correlation structure. But the optimal amount of variance inflation might well depend on the dimension and precise nature of the target distribution, the type of sampler used

(multivariate versus univariate, Metropolis versus Hastings, etc.), or any number of other factors.

Explore these issues in the context of the flour beetle mortality data in Table 3.3 by resetting Σ =. cΣ for c = 1 (candidate variance matched to the target) and c = 4 (candidate standard deviations twice those in the target) using

(a) the multivariate Hastings algorithm in part

(a) of problem 12.

(b) the univariate Metropolis algorithm in part

(b) of problem 12.

(c) the multivariate Metropolis algorithm originally used in Example 3.7.

Evaluate and compare performance using acceptance rates and lag 1 sample autocorrelations. Do your results offer any additions (or corrections) to the “folklore”?