Derivation of Similarity Function

Posterior probability is:

$$p(T\vert Y,S,Q) \propto \int p(Y\vert T,S,\beta,\alpha) p(T) p(\alpha \vert \lambda, Q) p(\lambda) p(\sigma) \, d\sigma \, d\lambda \, d\alpha$$

$$\propto p(T) \int \left( \frac{\beta}{2 \pi} \right)^{N/2} \exp\left( \fr... ...p(\alpha \vert \lambda, Q) p(\lambda) p(\beta) \, d\beta \, d\lambda \, d\alpha$$

Note that $Q$ is considered a known parameter and not marginalised.