 Solution

Start from any $(a^{(0)},b^{(0)})$. For independent draws $G_{1,k},G_{2,k}$ from the standard normal distribution at each sweep, the Gibbs sampler can be implemented asThe second update must use the newly drawn first coordinate. The transform in part (a) supplies the needed independent Gaussian draws. Each conditional update preserves the joint posterior, so their composition does too.

Convergence is particularly transparent here. Centering at the posterior mean givesPositive definiteness gives $C^2/(AD)<1$, so this is a stable Gaussian autoregression. The full sampler has the desired posterior as its limiting invariant law. This is the linear contraction of a two-coordinate Gaussian Gibbs sweep.

For a posterior-integrable function $\varphi$, its posterior expectation is estimated after a burn-in $K$ byThe ergodic theorem justifies this Monte Carlo average. The retained values of $\varphi$ also approximate its posterior distribution and quantiles. If a Bayesian point estimate is requested under squared-error loss, the posterior mean is the appropriate estimate, when its needed moments exist. An arbitrary function need not have a finite posterior mean; integrability must be assumed for the displayed target. Successive Gibbs draws are correlated, so uncertainty in the Monte Carlo average should use chain-aware error estimates rather than treating them as iid.