Suppose we fit a Bayesian logistic regression model of the form
$$Y_i \sim Bernoulli(p_i)$$
$$logit(p_i) = \beta_0 + \beta_1*x + \alpha_{j[i]}$$
$$\alpha_j \sim N(0,\sigma_\alpha^2)$$
$$\beta_i \sim N(0,1000)$$
and we obtain samples from MCMC of
$$(\beta_0^{(s)},\beta_1^{(s)},\alpha_j^{(s)})$$.
Suppose we wish to calculate the 95% Credible Interval for cluster 1.
Is it equivalent to take the quantile of
$$logit^{-1}(\beta_0^{(s)} +\beta_1^{(s)}*x + \alpha_1^{(s)}) , \ \forall s$$
or take
$$logit^{-1}(\beta_0^{.025/.975} + beta_1^{.025/975}x + \alpha_1^{.025/.975})$$
That is, can we take the quantiles of the coefficients first?
