\begin{equation}\label{eq:aag.log_rho.lognormal}
 \log \rho(\pi) = \mu + \frac{\sigma^2}{2} - \log(2 \pi) + \log \left( 1 + \mathrm{erf} \left( \frac{\sigma^2}{\sqrt 2} - \mathrm{erf}^{-1}(1 - 2\pi) \right) \right).
\end{equation}
