I know 3 methods to do parameter estimation, ML, MAP and Bayes approach. And for MAP and Bayes approach, we need to pick priors for parameters, right?
Say I have this model $p(x|\alpha,\beta)$, in which $\alpha,\beta$ are parameters, in order to do the estimation using MAP or Bayes, I read in the book that we'd better pick a conjugate prior $p(\alpha,\beta)$, which is a joint probability of $\alpha,\beta$, right?
I have 2 questions:
Do we have other choices picking the prior other than this conjugate one?
Can we pick priors for $\alpha$ and $\beta$ respectively like $p(\alpha)$ and $p(\beta)$, other than put them together in a joint one?