Figure A1.

(Left) The i-th component of the conditional posterior $p({\theta }_i\mid x_i)$ optimized with respect to the shape parameter $\beta$ so that ${\tilde{\theta }}_{i,\max }^{\left(\mathrm{G}\right)}\approx c{\tilde{\theta }}_{i,\max }^{\left(\mathrm{IG}\right)}$ close to the point of linearization $x_i^2=\alpha {\theta }_0/(\beta -1)$ with some constant c (Appendix A). The gray curves correspond to IG and the black ones to G hyperprior. The solid, dashed and dash-dotted curves have been obtained with $\alpha =100$, $\alpha =10$ and $\alpha =1000$, respectively. In each case the scale parameter is ${\theta }_0=1$. The maximizers match approximately, i.e., $c\approx 1$, when $\alpha =100$ (solid) and the optimizer is $\beta \approx 3$. (Center) The relative scale deviation $|c-1|/c$ w.r.t. $c=1$ as a function of the optimizer $\beta$, vanishing when $\beta \approx 3$. (Right) The optimizer $\beta$ as the function of $\alpha$.