\right],$$ where $F_{\sigma^2}(\cdot; a , b )$ is the CDF of the IG distribution. The minimisation cannot be done analytically so numerical optimisation is applied.

The expert would may find it hard to visualise her judgements from the fitted IG distribution since $\sigma^2$ is not directly observable. Therefore, we suggest showing the expert two density plots of the population distribution with a fixed mean and the two calculated quantiles of the variance, $ \mbox{N} (\hat{m}, {\sigma}^2_{(0.05)})$ and $\mbox{N} (\hat{m}, {\sigma}^2_{(0.95)})$. The expert visualises the variation in the distribution of $X_i$ based on her judgements about $\theta$, and then confirms whether the fitted IG distribution is a reasonable representation of her opinion about $\sigma^2$.

To summarise, the expert is asked to make six judgements, as set out in Table \[tab:elicited-questions\].

    ***Notation***    ***Elicitation Question / Definition***
  ------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
          $L$         What is your lower plausible bound for the quantity of interest where is it mostly unlikely for any $X_i$ to take values less than $L$?
          $U$         What is your upper plausible bound for the quantity of interest where is it mostly unlikely for any $X_i$ to take values more than $U$?
   $\mu_{\alpha_1}$   What is your $\alpha_1$ quantile of the population mean?
   $\mu_{\alpha_2}$   What 