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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?

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Yes, you can use a normal prior. Conjugacy is great because the posterior can be written down pen and paper. But there is nothing stopping you from specifying a normal as the prior for the probability (given you constrain it to be between 0 and 1). The catch is that you now have to use numerical methods to get the posterior.

Would you like an example using Stan, the popular open source language for fitting Bayesian models?

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  • $\begingroup$ actually I want to know how the marginal probability calculate by numerical method $\endgroup$ – Linear Algebra fans Jun 12 '20 at 11:19
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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.

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  • $\begingroup$ Is it ok to use any combination of distribution in MAP? $\endgroup$ – Linear Algebra fans Jun 12 '20 at 11:36
  • $\begingroup$ From a Bayesian perspective, any prior is valid, since the entire analysis is relative to the choice of the prior. $\endgroup$ – Xi'an Jun 12 '20 at 11:44

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