I am trying going through an exercise, I don't understand how the information provided in the text below transitions into the parameters displayed in the beta priors. How are these informative priors generated?
The expert starts 90% shure that the expected survival time in absence of the contaminant is less then 40 and 90% shure that it is greater then 10.
This is the information on which we base the prior to $\beta_0$ because the absence of the contaminent means we have a $conc_i = 0$ and thus $\beta_1$ does not matter. We look for a normal distribution with 10% of the mass below log(10) and 10% of the mass above log(40). As the normal distribution is symmetric around the mean, the mean of that standard deviation will be in the middle between log(10) and log(40), so
(log(10) + log(40))/2 = 2.9957. (Your screenshot says the mean was
Now all we need is a standard deviation with
qnorm(c(.1, .9), mean = 2.9957, sd = x) == c(log(10), log(40)) == c(2.302585, 3.688879).
Now there will be a number of ways to come to a standard deviation. For example a graphical approach:
curve(qnorm(.1, mean = 3, sd = x), ylim = c(1.5, 4.5), ylab = expression(log(lambda)), xlab = expression(sigma),col = "blue") curve(qnorm(.9, mean = 3, sd = x), add = TRUE, col = "blue") abline(h= c(log(10), log(40)), lty = 3, col = "grey") abline(v = .54, lty = 2, col = "grey")
Let's try the
.54 from your book:
> qnorm(c(.1, .9), mean = 2.9957, sd = .54)  2.303662 3.687738
Close enough. I will leave the $\beta_1$ for you to try because after all this is a self-study question, even if it does not have a self-study tag.