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I am reading about Bayesian inference. One book (DeGroot) discusses how different prior distributions can change the posterior distribution. Prior distributions are assumptions based on the statistician's beliefs, however, and I did not see any discussion on how to deduce what the prior distribution actually is. Are the observed data distribution and assumed prior distribution sufficient to find the true prior distribution?

Another book, Tsitsiklis, speaks of the MAP (Maximum A Posteriori) rule for when we want to find an estimator of $\theta$, a scalar value. The MAP rule finds the estimator of $\theta$ which maximizes the conditional probability $\Pr(\theta|x)$, or in Tsitsiklis' words, "maximizes the posterior distribution over all possible values of $\theta$. So to find the true prior distribution, how should we proceed? Does it make sense to follow the MAP method, that is, to find a prior distribution which maximizes the posterior distribution over all possible values of $\theta$?

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The prior distribution has nothing to do with either the data or the posterior estimator. The entire purpose of the prior is to express the client's state of uncertainty about unknown parameters before the observation of data. In practice, one could evoke it by a series of questions comparing the client's ideas about the parameter to known experiments.

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