 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2021/3/c/iii/solution

p

and

β

rather than judging the chain solely by its length:

Trace plots of each parameter and the log-posterior against iteration, including early iterations and later windows. Look for initial drift, long flat stretches, abrupt switches and sustained changes in spread. Several independent chains started at dispersed points should explore the same region after burn-in.
Plots of cumulative sample means and selected sample quantiles against iteration. Compare chains and successive long blocks; persistent movement indicates that the estimated posterior moments or tails have not stabilized.
Autocorrelation function plots for both parameters after tentative burn-in. Slow decay reveals strong serial dependence and a small effective sample size of a Markov chain, even when $1 0^{6}$ iterations have been generated.
Overlaid histograms or kernel density estimation curves from successive retained blocks and from different chains. Their agreement provides evidence of stable marginal posterior distributions.
A scatter plot of retained $(p, β)$ pairs, colored by time block if helpful. Strong dependence or separate clusters can expose slow joint exploration that marginal plots conceal.

A million iterations and stable-looking plots do not prove convergence. Agreement across dispersed chains, exploration of all visible modes and adequate effective sample size of a Markov chain provide stronger evidence. Estimate the Monte Carlo standard error using the chain's autocorrelation; the independent random variables formula from 3(b)(ii) does not apply directly to these dependent draws.

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