and in making the produce more robust to vaguely expressed judgements.

Given the relationship between $\theta$ and $\sigma^2$ in equation (\[equ:sigma-quantile\]), we consider relative changes in $\sigma$ as $\theta$ varies and so consider $\log \sigma$ as a function of $\theta$: $$\log \sigma = \log c - \Phi^{-1}(\theta + 0.5).$$ We plot this relationship (ignoring the $\log c$ term) in Figure \[fig:theta-sigma\]. The gradient is fairly constant and at its minimum over the range $[0.2, 0.45]$, so to make the procedure as robust as possible it would be desirable for the expert to report her quantiles for $\theta$ outside this interval. In particular, if the expert reports $\theta_{(0.05)}$ of say 0.05 or 0.1, the implied 95th percentile for $\sigma$ could change more appreciably given minor adjustments to $\theta_{(0.05)}$. In this case, the facilitator could consider repeating the process with a larger value of $c$.

\[fig:theta-sigma\]

(0,0) rectangle (252.94,252.94);

( 49.20, 61.20) rectangle (227.75,203.75);

( 55.81,198.47) – ( 55.98,197.85) – ( 56.14,197.24) – ( 56.31,196.64) – ( 56.47,196.05) – ( 56.64,195.48) – ( 56.81,194.91) – ( 56.97,194.35) – ( 57.14,193.81) – ( 57.30,193.27) – ( 57.47,192.74) – ( 57.63,192.22) – ( 57.80,191.70) – ( 57.96,191.20) – ( 58.13,190.70) – ( 58.30,190.21) – 