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I am trying to see what would be an appropriate prior distribution for $\theta$, the average GPA of all college students who graduate in a particular year Y from university U. Assume also that from past data we know that usually about $50\%$ of graduating college students from U obtain a grade of $3.0/4.0$ or higher.

What would be a meaningful prior distribution for $\theta$? Given that we are not completely clueless about the past distribution of $\theta$, it would be better not to put a uniform prior on $\theta$. I was thinking of putting a normal prior distribution on $\theta$ but this seems problematic too, since the support of a normal distribution is $\Bbb R$ while the support of the prior for $\theta$ is $[0,4]$. IS there some natural choice for the prior distribution given the above?

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    $\begingroup$ if you really want to use the normal you can put a normal prior on a transformed version of GPA (so that its on $\mathbb{R}$) e.g. $\text{GPA_transform} = logit(GPA/4)$ $\endgroup$
    – bdeonovic
    Nov 28 '16 at 14:36
  • $\begingroup$ Do you have the actual data or do you want a conservative choice in the general case where you don't know the data? In the former case you could post a histogram. $\endgroup$ Nov 28 '16 at 19:29
  • $\begingroup$ @user7019377 I don't have any data (except for the information I am already providing). I am just trying to figure out a reasonable prior for this kind of a parameter in a general scenario, nothing specific given. $\endgroup$ Nov 28 '16 at 19:31
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I would say that a natural choice of a distribution for a bounded real-valued variable is beta distribution. Standard beta distribution is bounded in $[0,1]$, but you can introduce additional parameters for lower and upper bound. Moreover, you can re-parametrize beta in terms of mean ($\mu = \tfrac{\alpha}{\alpha+\beta}$) and precision ($\phi = \alpha+\beta$),

$$ f(x) = \frac{1}{\mathrm{B}(\mu\phi, \,(1-\mu)\phi)} x^{\mu\phi-1} (1-x)^{(1-\mu)\phi-1} $$

what makes choosing the appropriate parameters easier.

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