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

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

Codex  0 2026-10-20

TODO: Complete the data-specific numerical results and diagnostics when the original Iris_data.csv is available.

Use the random-walk Metropolis algorithm with proposal

μ^{'} = μ + ε

, where

ε \sim N (0, 1)

. The proposal is symmetric, so for a nonnegative proposal the acceptance probability is

a (μ, μ^{'}) = min {1, exp [ℓ_{p} (μ^{'}) - ℓ_{p} (μ)]}, ℓ_{p} (u) = - \frac{1}{2} \sum_{i} (x_{i} - u)^{2} - \frac{u}{3} .

(1)

Negative proposals are rejected. They are not redrawn until positive, since that would change the proposal kernel and require a different acceptance ratio. On rejection, store the current state again; keeping accepted moves alone would bias the empirical posterior distribution.

This R continuation makes exactly 10,000 proposals, starting at

μ_{0} = 4

set.seed(20240528)
B <- 10000L
mu <- numeric(B + 1L)
mu[1L] <- 4
logpost <- function(u) {
  if (!is.finite(u) || u < 0) return(-Inf)
  loglik(u) - u/3
}
lp <- logpost(mu[1L])
accepted <- 0L
for (i in seq_len(B)) {
  proposal <- rnorm(1L, mean = mu[i], sd = 1)
  lp_new <- logpost(proposal)
  if (log(runif(1L)) < min(0, lp_new - lp)) {
    mu[i + 1L] <- proposal
    lp <- lp_new
    accepted <- accepted + 1L
  } else {
    mu[i + 1L] <- mu[i]
  }
}
cat("Acceptance fraction =", accepted/B, "\n")
# A provisional discard; assess it with the trace in part (e).
burn <- 1000L
post <- mu[(burn + 2L):(B + 1L)]
m <- xbar - 1/(3*n)
s <- 1/sqrt(n)
posterior_density <- function(u) {
  ans <- numeric(length(u))
  ok <- u >= 0
  ans[ok] <- exp(dnorm(u[ok], m, s, log = TRUE) -
                   pnorm(m/s, log.p = TRUE))
  ans
}
png("q2-posterior.png", width = 900, height = 600)
hist(post, breaks = 40, probability = TRUE, xlab = expression(mu),
     main = "Retained Metropolis draws and exact posterior")
u <- seq(0, max(post) + 3*s, length.out = 4001L)
lines(u, posterior_density(u), col = "firebrick", lwd = 2)
dev.off()

The exact truncated normal distribution density overlays the histogram as a check; it does not replace the requested Markov chain Monte Carlo procedure. The 9,000 retained states exclude the initial state and the first 1,000 transitions. The particular histogram and acceptance fraction remain dependent on the missing measurements.

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