# Conditional distribution for Neal's funnel

A simple version of Neal's funnel is:
$$
\begin{align*}
Y &\sim N(0,\sigma^2) \\
X &\sim N(0,\exp(y)^2)
\end{align*}
$$

The corresponding densities are
$$
\begin{align*}
p(y) &\propto \exp(-\tfrac{1}{2} \sigma^{-2} y^2) \\
p(x|y) &\propto \exp(-y) \exp( -\tfrac{1}{2} \exp(y)^{-2} x^2)
\end{align*}
$$

So the joint density is
$$
p(x,y) \propto \exp(-\tfrac{1}{2} \sigma^{-2} y^2 -y -\tfrac{1}{2} \exp(y)^{-2} x^2)
$$
Thus the conditional
$$
p(y|x) \propto \exp(-\tfrac{1}{2} \sigma^{-2} y^2 -y -\tfrac{1}{2} \exp(y)^{-2} x^2)
$$
For large $y$ this is asymptotic to $\exp(-\tfrac{1}{2} \sigma^{-2} y^2)$.

So: **under Neal's funnel, $Y|X=x$ has a Gaussian tail**.