Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 1 / h / Solution

Fit two hierarchical Bayesian models to the same observations, one with the bounded uniform distribution sampling density and the other with an exponential distribution density. Specify rate versus mean in the exponential model and assign appropriate, scientifically comparable proper prior distributions; that parameter is no longer a literal maximum size. Obtain posterior distributions, for example by Markov chain Monte Carlo, and check convergence.

At the same observational level in both models use the Bayesian deviance $D(\varphi )=-2\log p(\mathbf{y}\mid \varphi )$, excluding prior densities. Retain the same likelihood constants and consistently either condition on breed effects or integrate them out. Estimate $\overline{D}=\mathbb{E}[D(\varphi )\mid \mathbf{y}]$, evaluate $D(\overline{\varphi })$ at the posterior mean, and compute the effective parameter count in DIC:Smaller DIC favours the fit–complexity tradeoff. Supplement it with posterior predictive checks; the uniform model's parameter-dependent support and the hierarchical structure make the criterion a diagnostic rather than an automatic definitive decision.