Past exam of the mathematics course of the University of Cambridge 2008 iii Paper 44 4 d Solution Created 2026-10-03 Updated 2026-10-14
Model B replaces independent tree parameters by multivariate normal random effects. It permits correlated asymptote, timing and slope variation and supplies partial pooling, a useful improvement if the five trees can be regarded as exchangeable.
The Wishart distribution prior is on the precision matrix , not the covariance matrix. In the WinBUGS convention, dimension three and degrees of freedom three give a proper but very dispersed prior. The induced inverse-Wishart distribution has no finite mean at those degrees of freedom. Thus this is not a neutral choice of covariance prior, and estimating six covariance components from only five trees is weakly informed. The broad mean priors and possible negative growth slopes also remain. Check prior-predictive curves and sensitivity, and consider separate, interpretable scale and correlation priors.
The resulting posterior means of the second and third parameters are quite similar across trees, motivating a simpler shared-shape model; their uncertainty means this similarity is evidence to explore, rather than a proof of exact equality.