Estimate the convolution model, together with predictors \(X_{1}-X_{4}\) for the English crime rate data, namely

\[\log \left(\theta_{i}\right)=\alpha+X_{i} \beta+\epsilon_{i}+u_{i}\]

where the \(\epsilon_{i}\) are \(\operatorname{ICAR}(1)\) and the \(u_{i}\) are iid. Assess convergence in \(V_{\epsilon} /\left(V_{u}+V_{\epsilon}\right)\), and the impact that including predictors has on the relative importance of spatial to iid random effects. Compare the LPML with that obtained for the predictor model combined with the Leroux et al. (1999) spatial prior.