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I think this is a fairly beginner bayesian analysis question.

I have a Beta Posterior with $\alpha = .32$ and $\beta = 1.35$ (estimated using MCMC), that describes a probability.

My question is: what is the best way to take a point estimate of the probability? My first thought for a posterior is to take the mode, since this will be the most probable value of my beta distributed random variable. However, with the above parameterization my pdf goes to infinity as p goes to 0 . enter image description here

And indeed, wikipedia suggests that having $\alpha, \beta < 0$ mean there isn't a non-infinite maximum of this curve.

I don't think $p=0$ is the best answer, so is it valid to take the mean?

I don't have much experience with Bayesian analysis, so any advice/help/links to similar questions are appreciated!

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  • $\begingroup$ If you must choose a point estimate, then it should be informed by your loss function: if that is the square of the error then its expectation is minimised by the mean of the posterior distribution; if it is the absolute error then its expectation is minimised by the median of the posterior distribution, and so on. The mode of the density has little to recommend it in this sense (the mode of a discrete posterior might work if your loss function was constant for any non-zero error) $\endgroup$
    – Henry
    Commented May 11, 2020 at 23:12

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If you could take the mode (you can't here), its called MAP (Maximum a Posteriori) estimate. It's a common point estimator (may not be the best sometimes). If you take the expected value/mean it's called conditional expectation given data, $\mathbb E[X|\mathcal D]$ or the posterior mean, which is a commonly used point estimator as well.

The advantage of methods similar to MCMC is having the posterior distribution available, i.e. you can find any defined moment you like, assess the variance, and have confidence intervals etc. Having a point estimate for the mean is also possible, but undervalues the importance of the posterior.

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The mode probably isn't a great statistic to take in that case. How about the median or the mean (particularly since you aren't affected by extremely large values, as you are bounded between 0 and 1)?

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