\begin{align}
    \mathcal{Z}^0(y, \mu, \sigma) & :=\int \frac{P_y^0(y \mid x) \mathrm{d} x}{\sqrt{2 \pi \sigma}} \mathrm{e}^{-\frac{(x-\mu)^2}{2 \sigma}} = \frac{1}{\sqrt{2 \pi (\zeta^2 + \sigma)}} \exp \left(-\frac{(y-\mu)^2}{2 (\zeta^2 + \sigma)}\right)
\end{align}
