= Solution
\b[TODO: Complete the data-specific numerical results and diagnostics when the original `Iris_data.csv` is available.]
Plot all 10,001 stored states, including the initial value, and indicate the provisional <burn-in> boundary. Also display the retained chain's <autocorrelation function>:
``
png("q2-trace.png", width = 1000, height = 700)
par(mfrow = c(2, 1), mar = c(4, 4.5, 2.5, 1))
plot(0:B, mu, type = "l", xlab = "Iteration", ylab = expression(mu),
main = "Complete Metropolis trace")
abline(v = burn, col = "firebrick", lty = 2)
abline(h = m, col = "steelblue", lty = 3)
acf(post, main = "Autocorrelation after provisional burn-in")
dev.off()
# Check sensitivity to the chosen discard and to the retained time window.
for (discard in c(500L, 1000L, 2000L)) {
kept <- mu[(discard + 2L):(B + 1L)]
cat("Discard", discard, ": posterior mean", mean(kept), "\n")
}
cat("Means of retained halves:", mean(post[1:4500]),
mean(post[4501:9000]), "\n")
``
A satisfactory <trace plot> should show an initial transient, if any, followed by repeated exploration of one stable range with no continuing drift. Repeated values are expected because rejections are stored. If $n$ is large, the prescribed proposal standard deviation one can be much wider than the <posterior> scale $1/\sqrt n$, giving low acceptance and conspicuous plateaus. A flat-looking trace is therefore not by itself evidence of good exploration.
Compare the retained histogram with the exact <posterior density>, inspect <autocorrelation>, compare separate retained windows, and assess sensitivity to the discarded count. Agreement supports the adequacy of this run; neither 10,000 iterations nor a chosen <burn-in> of 1,000 proves convergence. The <effective sample size of a Markov chain> can be much smaller than the number of stored states.
\b[The actual trace cannot be diagnosed without `Iris_data.csv`.] The code produces the required trace when the measurements are available; the comments here specify what to examine rather than asserting an unseen chain has converged.
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