I am sure the mean of \theta_i is not equal to 0.5. So \alpha \neq \beta. Now, how to determine the value of hyperparameters \alpha and \beta as noninformative priors?
\theta \sim \beta(\alpha,\beta) E[\theta]=\alpha/(\alpha+\beta)
Set \alpha=\mu*s, \beta=(1-\mu)*s with \mu mean and s scaling parameter.
Then E[\theta]=\mu s / ((1-\mu)s+\mu s)=\mu
In Stan
mu = inv_logit(something);
y ~ beta_binomial(n, mu * phi, (1-mu) * phi);
Yes. I know reparameterize two shape parameters, \alpha and \beta, as mean parameter \phi and total count parameter,\lambda, is a normal approach. The Stan user guide gives the following noninformative prior:
phi ~ beta(1, 1);
lambda ~ pareto(0.1, 1.5);
I tried multiple noninformative priors and found Haldane’s prior Beta (0,0) can not run.
How to solve the issue? Any help is appreciated!