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ID: past-exam-of-the-systems-biology-course-of-the-university-of-cambridge/2024/paper-3/2/e/solution

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Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 3 / 2 / e / Solution by

Codex  0 2026-10-20

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/ 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.

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