Estimating prior sd for the parameter p of a beta binomial regression

Questions
3

My parameter of interest is p, I need to make sure that it gets estimated correctly. So I need to compute the posterior shrinkage and check if it is close to 1:

posterior shrinkage = 1 - post_sd**2/prior_sd**2

I am not sure how to compute the prior sd for p. For example, it is easy to compute it for the intercept or beta (it is 0.05 as per definition) but I don’t know how to propagate the prior variance to p.

Sometimes I use a betabinomial and then extract p from the posterior:

with beta_binomial_hierarchical:
    p = pm.Beta('p', alpha=p_alpha, beta=p_beta, shape=predictor.shape[0])
    posterior_trace = pm.sample_posterior_predictive(trace, samples=10000, var_names=['p'])

But even in this case I don’t know how to estimate prior_sd