# Fitted values and diagnostics in MCMC model

I'm fitting some GLM models with different link functions (logit, probit and cloglog) with JAGS package. I have no experience at all with MCMC based models then I have three main doubts:

1) After I fitting the model, I will have the posterior estimates for the parameters. I can use this values to create a function to generate predictions and residuals like $$logit(y)=X\beta\Rightarrow \hat{Y}=\frac{e^{X\hat{\beta}}}{1+e^{X\hat{\beta}}}$$ $$e=Y-\hat{Y}$$ or it is not valid?

2) How I can compare the same model with different link functions to choose the best?

3) Why are most diagnoses in Bayesian models via MCMC not using residual analysis? How do you evaluate the predictions in Bayesian models in general?