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?