Given a distribution $\{p(X|\theta),\theta\in\Theta\}$:

  • how do you show the existence/non-existence of its conjugate prior?

  • what are some general ways/principles to construct/find its conjugate prior, if there is one?

  • If it exists, how do you show it is unique or not?

  • If not unique, which one is the most "canonical/classical" (in your definition) one among all?

  • $\begingroup$ If we define a family of prior distributions $P$ to be conjugate to a likelihood if the posterior also lies in $P$, then the family of all distributions is trivially conjugate to any distribution (i.e. a conjugate prior always exists). And when a simple conjugate family exists, mixtures of this family are also conjugate (i.e. it is not unique unless it's trivial). $\endgroup$ Commented Mar 24, 2014 at 12:41