Writing $\varphi(u;m,v)$ for the $N(m,v)$ density, the conditional-independence structure of the <latent-variable model> gives
$$
p(y_i,x_i,\eta_i,\xi_i\mid\alpha,\beta,\sigma^2,\mu,\tau^2)
=\varphi(\xi_i;\mu,\tau^2)
\varphi(\eta_i;\alpha+\beta\xi_i,\sigma^2)
\varphi(x_i;\xi_i,\sigma_x^2)
\varphi(y_i;\eta_i,\sigma_y^2).
$$

Solution

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

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A latent-variable model represents observed data through unobserved random quantities whose marginalization induces the observed distribution.


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