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A trace of a Markov chain Monte Carlo parameter helps distinguish slow movement, long rejection plateaus, and movement between separated regions. Very small random-walk proposals give high acceptance but slow diffusion; very large proposals produce low acceptance and repeated states. Record acceptance rates alongside trace plots and effective sample size of a Markov chain before deciding which scale to change. Correlated posterior directions often benefit from block proposals with a covariance fitted during a preliminary tuning run. A trace alone cannot prove convergence or identify the unique cause of poor movement; dispersed chains and stationary summaries provide additional evidence. Freeze tuning before the main sampling phase, or use adaptation satisfying an appropriate ergodicity theorem.

Convergence means that the chain's distribution approaches the intended posterior, independently of its initial state, under the relevant ergodicity assumptions. It is separate from obtaining enough effectively independent draws for a precise answer. Start several chains at dispersed plausible points, examine trace plots and stability of posterior summaries, compare between-chain and within-chain variation, and inspect autocorrelation and effective sample size of a Markov chain. Discard an initial transient when appropriate and report Monte Carlo uncertainty.

No finite diagnostic proves convergence. Agreement among chains started in the same basin can miss another mode. If different starting points give incompatible summaries, extend runs or improve proposals rather than merely deleting more observations. Extra thinning does not repair poor exploration; retaining all post-transient draws usually preserves more information for estimating uncertainty.

The Metropolis–Hastings algorithm is useful when a target posterior can be evaluated up to a normalizing constant but direct independent sampling or simple conditional sampling is unavailable. It constructs a Markov chain with that posterior as invariant law. After adequate convergence, averages of functions of the chain estimate posterior expectations, and the retained draws approximate marginal distributions, quantiles and credible intervals.

For the random-walk Metropolis algorithm, at current $\theta$ propose ${\theta }^{\prime }=\theta +Z$, where $Z$ has a symmetric distribution, often $N(0,\Sigma )$. With a symmetric proposal the acceptance probability isAccept the proposal when an independent uniform is below this probability; otherwise retain the current value. For a nonsymmetric proposal $q({\theta }^{\prime }\mid \theta )$, include $q(\theta \mid {\theta }^{\prime })/q({\theta }^{\prime }\mid \theta )$ in the ratio. Detailed balance follows because the accepted flow is the minimum of the two proposed directional flows; rejection supplies the remaining self-transition probability. Irreducibility, appropriate recurrence and aperiodicity are also needed for convergence.

In the displayed trace plots, the first parameter moves in small increments with pronounced long-term wandering and a shift from its early region to a later region. This is consistent with an undersized proposal or a strongly correlated slow direction. Measure its acceptance rate: if it is very high, increase that coordinate's proposal variance. The second parameter moves rapidly within regions but changes its level around iteration 4500; that is evidence against treating the full run as a settled stationary sample, and it suggests investigating dependence with the first parameter or separated posterior regions. Its local scale may already be reasonable, so a blanket reduction of all proposal variances would be unwarranted. The third parameter has conspicuous long plateaus, indicating many repeated states: if the recorded acceptance is low, reduce its proposal variance; also check constraints and dependence before deciding that scale alone is the cause.

Rerun with coordinate-specific scale changes, inspect joint dependence, consider a covariance-based block proposal, and run longer chains from dispersed starting points. The MCMC trace diagnosis and proposal tuning should include actual acceptance rates and effective sample sizes; the three marginal trace plots do not by themselves identify correlations or prove convergence. Further practical details are in the three subparts.

Use several complementary Markov chain Monte Carlo convergence diagnostics, monitoring both $p$ and $\beta$ rather than judging the chain solely by its length:

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.

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.

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.

A trace plot displays an observable along successive iterations of a Markov chain Monte Carlo algorithm. Its horizontal axis is iteration and its vertical axis is the observed parameter or function value. Trends suggest an initial transient or slow exploration, separated regimes suggest transitions between posterior regions, and exact plateaus show repeated states. The plot should be read with acceptance statistics, autocorrelation and multiple starting states; its visual appearance alone cannot prove stationarity or identify the unique cause of poor mixing.