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Is maximum a posteriori estimation some kind of Bayesian Estimation?

If yes, can you point out other Bayesian estimators?

Edit:

So I've come to know the following (don't know if they are correct):

MAP finds the mode of the posterior, and Bayesian Estimator finds the distribution of that posterior with respect to the estimating parameters. And in order for Bayesian Estimator to find the optimal parameters, some particular risk function (like Mean Squared Error, 0/1 error etc.) must be defined.

So it seems these two estimators have nothing in common except they both incorporate prior into their estimation. Or is it possible that MAP is some special case of Bayesian Estimator with some particular risk function.

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    $\begingroup$ Welcome to CV. It would be helpful if you provided some additional details about your question as it's a bit vague now. You might refer to some of the core texts of Bayesian analysis to firm up your query such as Gelman, Carlin, et al's Bayesian Data Analysis which is out in multiple editions. $\endgroup$
    – Mike Hunter
    Feb 4, 2016 at 17:42

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MAP finds the mode of the posterior, while full Bayes characterizes the entire distribution -- the probability assigned to each element, whence we can know all of its moments and so on. Chapter 4 of Gelman's Bayesian Data Analysis 3rd ed develops this point in more detail.

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  • $\begingroup$ Is it only the mode or also other statistical parameters such as mean, variance? $\endgroup$
    – Outcast
    Oct 30, 2018 at 12:18
  • $\begingroup$ @Penseur By definition, MAP only estimates the mode because the mode is where the probability mass is maximized. Refer to BDA for more information. $\endgroup$
    – Sycorax ♦
    Feb 20, 2020 at 15:05

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