# The prior in MAP and Bayesian interference

We can use a Normal distribution as a prior when handling a Normal distribution as likelihood in Bayesian inference

However if we want to do MAP given a Bernoulli as likelihood can we use Normal distribution as a prior which ignore the conjugate rule ? or we only allow to use Beta Prior?

Assuming you observe a Bernoulli variate $$X\sim\mathcal B(p)$$, you cannot use a Normal prior in a strict sense on $$p$$ since $$p\in(0,1)$$. Unless you set a Normal prior on the unconstrained parameter$$\theta=\log\frac{p}{1-p}$$ (remember that a prior is associated with a specific parameterisation of the likelihood). In both cases the Normal priors are not conjugate, if this is of importance. (It should not be.)
Note also that any prior distribution on $$(0,1)$$ is acceptable as a prior, especially if the only goal is in deriving the MAP estimate, since this is not truly a Bayesian derivation.