Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 2

Writing $\phi (u;m,v)$ for the $N(m,v)$ density, the conditional-independence structure of the latent-variable model gives

First integrate out ${\eta }_i$: conditional on ${\xi }_i$, $y_i\sim N(\alpha +\beta {\xi }_i,s^2)$ with $s^2={\sigma }^2+{\sigma }_y^2$. Since $(x_i,y_i)$ is an affine transformation of independent normal variables, it is bivariate normal with meanand covarianceIts determinant simplifies toFor $r_i=(x_i-\mu ,y_i-\alpha -\beta \mu {)}^T$, the observed-data likelihood is therefore

Take flat priors on $\alpha ,\beta ,\mu$ and scale priors $\pi ({\sigma }^2,{\tau }^2)\propto ({\sigma }^2{\tau }^2{)}^{-1}$ on the positive variances. The posterior distribution is thenA random-walk Metropolis–Hastings algorithm can update $(\alpha ,\beta ,\mu ,\log {\sigma }^2,\log {\tau }^2)$ with a symmetric proposal and accept a proposed state $z^{\prime }$ from $z$ with probability $1\wedge \pi (z^{\prime }\mid \mathcal{D})/\pi (z\mid \mathcal{D})$, including the Jacobian if the target is represented in transformed coordinates. Its transition kernel satisfieswhich is detailed balance; hence the posterior is stationary.

The marginal predictor density isGaussian conditional expectation givesIntegrating this conditional distribution through the response model yields

When $\tau \gg {\sigma }_x$, $\kappa \to 1$, $v\to {\sigma }_x^2$, and $A\sim {\tau }^2$. HencewhereThe approximate maximum-likelihood estimators are