# Can my Bayesian prior reflect what the data should say rather than what it could say?

Can my Bayesian prior reflect what the data should say rather than what it could say? For example, assume I collect data where $Y_i$ is whether or not student $i$ passed the test and $X_i$ is whether or not the student studied for the test.

I want to consider the following model: $Y_i \sim Bern(\alpha + \beta X_i)$.

Now, it does not make sense that $\beta$, in realitiy, is negative. Studying for a test simply cannot reduce the probability of you passing. However, due to random chance, it is possible that the data shows that $\beta$ has an estimated negative effect.

For this reason, I would like to put a prior on $\beta$ so that, with probability 1, it is positive. My intuition tells me that this shouldn't be a problem and it is analogous to a restricted MLE in a frequentist framework. If the data shows an estimated negative effect, the restricted MLE will be wonky and have huge standard error. My intuition tells me it would be similar in a Bayesian framework, you would get a wonky posterior on $\beta$ with huge variance.

Is my intuition correct? Are there any dangers I am over-looking for having a prior that cannot be negative where the data can be?

Certainly you can have a prior for $\beta$ with support only over positive values and if the data suggest a negative value for $\beta$ then most of the posterior mass would be near zero, but positive which could actually result in a posterior that has a very little uncertainty (posterior variance).
One reason this may happen is that $X_i$ may not measure what you want it to measure. In this example, (I believe) $X_i$ is meant to measure preparedness for the exam, but perhaps the students who are most prepared do not need to "study for the exam". In which case, you would actually expect a negative value for $\beta$.