= 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> $D(\phi)=-2\log p(\mathbf y\mid\phi)$, excluding prior densities. Retain the same likelihood constants and consistently either condition on breed effects or integrate them out. Estimate $\overline D=\mathbb E[D(\phi)\mid\mathbf y]$, evaluate $D(\overline\phi)$ at the <posterior mean>, and compute the <effective parameter count in DIC>:
$$
p_D=\overline D-D(\overline\phi),\qquad
\boxed{\operatorname{DIC}=\overline D+p_D
=2\overline D-D(\overline\phi).}
$$
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.
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