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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?

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First off, with the current model there is some probability that the Bernoulli parameter is not between 0 and 1 and therefore you may want to constrain (usually via transformation) so that this parameter cannot be outside 0 and 1.

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).

Generally I suggest not using prior information to truncate distributions. Instead, I suggest constructing a prior that has most (but not all) of its mass on positive values. This way, if the data suggest that the value is truly negative, the data are allowed to overrule the prior.

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 $\beta$.

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This is a clear example of using informative prior, i.e. you have some prior beliefs about the data and you include it in the model as a prior. There is much more that can be done with informative priors, Spiegelhalter et al (2004) in their paper "Incorporating Bayesian Ideas into Health-Care Evaluation" review some of the usages, where choosing different priors could facilitate hypothesis testing or decision making using the data.

As about dangers, with exam scores it sometimes happens that the students that have higher abilities score lower on certain kinds of questions. For example, if there is a choice of an "obvious" or a complicated (but wrong) answer, it happens in practice that the more clever students choose the more complicated answer since the other one is "too obvious to be true". So for you this could mean that you could try a prior that gives a higher probability of positive values but allows negative values with a small probability. Finally, you could try different priors and compare the results (like Spiegelhalter et al. suggest).

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