Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 47 / 5 / b / i

A reversible-jump Markov chain Monte Carlo calculation compares normalized model-specific densities. The fixed-order posterior written only up to proportionality in part (a) omits normalizing factors that depend on $k$ and therefore cannot simply be canceled between dimensions. Moreover, the printed question does not specify the model-order prior distribution or the probabilities of selecting birth and death moves. Denote those by $p_k$, $b_k$ and $d_k$, respectively, retaining them in the answer.

Put $s={\sigma }^2$, $c=n^{-1}{\sum }_i{x}_i^{k+1}$, $v_i=x_i^{k+1}-c$, and $r=y-X_k\beta$. The centered polynomial birth move in reversible-jump sampling changes the fitted vector by $zv$, so its new residual is $r^{\prime }=r-zv$. ThereforeThe map from $({\beta }_0,\dots ,{\beta }_k,z)$ to $({\beta }_0^{\prime },\dots ,{\beta }_{k+1}^{\prime })$ has ones down the diagonal and a single additional entry $-c$ in the intercept row. Its Jacobian determinant is one.

Write $Q_k=(\beta -{\mu }_k{)}^T{\Sigma }_k^{-1}(\beta -{\mu }_k)$ and $Q_{k+1}=({\beta }^{\prime }-{\mu }_{k+1}{)}^T{\Sigma }_{k+1}^{-1}({\beta }^{\prime }-{\mu }_{k+1})$. The normalized multivariate normal distribution prior ratio isDivide also by the forward proposal density $q(z)=(2\pi {\sigma }_{\beta }^2{)}^{-1/2}{e}^{-z^2/(2{\sigma }_{\beta }^2)}$. The common inverse-gamma distribution prior for $s$ and the common normal-likelihood normalization cancel, since $s$ stays fixed. The forward acceptance ratio is consequentlyHere ${\sigma }_{\beta }>0$ is the proposal standard deviation. If equal model priors and symmetric birth/death selection are explicitly adopted, their ratios become one; they are not determined by the PDF. The expression assumes the same hyperparameters for the variance prior in both models, as in part (a). A dimension-dependent variance prior would contribute its own ratio.