# Elicit a Proper informative priors

I need help with eliciting,deriving and plotting two proper informative prior.

One with mean and standard deviation

Another with LQ, UQ and Median

Is the first one using a normal distribution while the second uses a scaled beta distribution? I'm not honestly very good at priors, so any help would be very appreciated.

• Your question may need a lot of work to be answerable. Eliciting priors from whom? Deriving in what sense? Proper prior in the sense that it integrates to one, which is the formal definition, or do you mean something more like a useful or sound prior? Feb 28 '19 at 14:24
• For prior #1, the way you present it makes it sound like the person proposing it was thinking of a normal distribution, because those are the two parameters that characterize a normal distribution. But that doesn't mean that a normal prior makes sense in the context. Feb 28 '19 at 14:24
• For prior #2, I assume you gave us the first, second, and third quartiles of the distribution. If you do the math, the first and second quartiles are pretty close to symmetric about the median, so in principle, you could represent that prior with a normal distribution as well. The thing is, that may or may not make sense in the context you're trying to use it. It may also be relevant how you got those two priors. Feb 28 '19 at 14:26
• Sorry, I should have included the context. The context is that in an drug experiment, patients with a condition are asked to choose between two drugs C(the control) and T(the treatment). The proportion population who prefer drug T is $\theta$. There are a sample of n patients which their response are independent given $\theta$. The question wants me to elicit those two prior distributions with those statements from experts as above Feb 28 '19 at 16:07
• A couple issues. If this is from a homework question, please tag it as self-study. See the link for the homework policy: stats.stackexchange.com/help/on-topic Feb 28 '19 at 16:21

I'm answering on the premise that this is a homework question. Disclosure: I'm not a Bayesian, but I'm familiar with some concepts.

From comments, you appear to want to elicit a prior that describes a fraction. You seem to be asking, what sort of prior is conceptually sound for this? You seem like you may be debating between a normal prior and a beta prior.

The normal distribution has support of -$$\infty$$ to $$\infty$$. So, in principle, assuming a normal prior could mean that you have fractions less than zero or over 1. That might not be substantively sound.

The beta distribution has support (0, 1), i.e. 0 to 1 exclusive of 0 and 1. If you Google, you should find that it's often used as a prior for fractions. It has two key parameters, $$\alpha$$ and $$\beta$$. If you had $$\alpha + \beta$$ trials and you got $$\alpha$$ successes, you can derive the appropriate parameters for a beta distribution expressing that as a probability.

Imagine you asked 10 patients for their preference, and 3 said they'd prefer the treatment, i.e. 30% of the sample prefer the treatment, but it's a small sample, so we're not sure the 'true' proportion is 30% (clearly I'm a frequentist). That's a beta distribution with $$\alpha = 3$$ and $$\beta = 7$$, and its cumulative distribution function looks like this:

If you had instead asked 1000 patients for their preference, and 300 said they'd prefer the treatment, you have $$\alpha = 300$$ and $$\beta = 700$$, you still think that 30% of patients prefer the treatment, but you're a lot more certain about that value, as illustrated by this very different CDF:

If you want to make similar plots, you can just go to Wolfram Alpha and tell it, for example, "plot a beta distribution with a = 300 and b = 700".

I'm not sure how you are eliciting priors from experts on this. In principle, if you asked a statistician who had studied this area for a prior, she might respond, "I think the probability of choosing T follows a beta distribution with $$\alpha = 3$$ and $$\beta = 7$$, and unlike Mr. Ng, I'm a real Bayesian." It doesn't seem likely you'd get this sort of response. I rather suspect you would be asking a bunch of experts, maybe 10, if they think that the average patient would prefer C or T. Thus, you would be able to parameterize your prior according to how many respond C or T. If they give a more complex response, then I guess you will have to ask them to choose one or the other (or bribe them, or threaten them, or whatever).

I'm not familiar with "scaled" beta distributions. However, I hope this gets you on the right track to think about how and why you might use the beta distribution as a prior.