What is the intuitive meaning of these statements in the context Bayesian prior and posterior?

Question

I am now familiar with the Bayesian thinking process of using a prior and then getting the posterior once we observe data using the prior.

I read the following statements which I am trying to get my intuition but unable so far. Any intuitive explanation will help.

A conjugate prior may always not lead to a better MAP estimate. Also, it is not clear what is a "better" MAP estimate

Why does a conjugate prior always not lead to a better MAP? I thought if you have a conjugate prior it is always easy to maximise the posterior integral.

Another downside of a choosing a conjugate distribution for the prior is that for some problems the conjugate may be inadequate.

In the above context what does inadequate actually mean?

Thanks a lot. Much appreciated.

Answer 1

Conjugate priors may not be the best model choice if they are chosen for convenience (i.e., ease of deriving the posterior). Your priority -- ideally -- should be to select a prior that best describes your belief. For example, suppose you have a Poisson likelihood and have a prior belief that the rate parameter is uniformly likely to be (1,7), then you should use a uniform prior. You should not use a Gamma(.) just because it would yield a Gamma posterior. This should answer both questions. If it doesn't, please let me know if you need any clarifications.