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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?

This is the model: enter image description here

enter image description here

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    $\begingroup$ Can you please include the parts below that line in the screenshot? $\endgroup$ – Arya McCarthy May 12 at 3:00
  • $\begingroup$ @AryaMcCarthy done, thank you. $\endgroup$ – frantic oreo May 12 at 6:24
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    $\begingroup$ Please type your question as text, do not just post a photograph or screenshot (see here). When you retype the question, add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ – kjetil b halvorsen May 12 at 7:46
  • $\begingroup$ @kjetilbhalvorsen Thanks, I would have typed the question however am not well versed in writing equations in Latex. $\endgroup$ – frantic oreo May 12 at 8:03
  • $\begingroup$ See math.meta.stackexchange.com/questions/5020/… $\endgroup$ – kjetil b halvorsen May 12 at 22:20
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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 3).

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")

enter image description here

Let's try the .54 from your book:

> qnorm(c(.1, .9), mean = 2.9957, sd = .54)
[1] 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.

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