Exercise 1: Eliciting an expert prior

In the context of the historical example, we want to elicitate a prior distribution from expert knowledge.

Now, let’s imagine we have 2 expert demographers, each giving their expert opinion about the value they think plausible for $\theta$ (the probability of a birth being a girl rather than a boy).

We asks each of them to give values for which the probability of $\theta$ being lower would be respectively 10%, 25%, 50%, 75%, and 90%. First expert says: $$P(\theta <0.2)=10\mathrm{\%},P(\theta <0.4)=25\mathrm{\%},P(\theta <0.5)=50\mathrm{\%},P(\theta <0.6)=75\mathrm{\%},\text{and }P(\theta <0.8)=90\mathrm{\%}$$ while the second expert says: $$P(\theta <0.5)=10\mathrm{\%},P(\theta <0.6)=25\mathrm{\%},P(\theta <0.7)=50\mathrm{\%},P(\theta <0.8)=75\mathrm{\%},\text{and }P(\theta <0.9)=90\mathrm{\%}$$

0. First, let’s load the SHELF R package:

   library(SHELF)

1. Using the fitdist() function from the SHELF package, estimate the parameter form the Beta distribution that best fit each of those elicitations (Protip: have a look at the package intro vignette here).

2. Plot those two Beta prior distributions, along with the “linear pooling” of their 2 curves using the plotfit() function from the SHELF package (Protip: use the lp = TRUE argument).

3. Derive a consensus Beta prior, by averaging each of both expert quantiles, and plot it.