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A bit, but in general, computing posterior distributions tends to very quickly lead to more complicated integrals. It just so happens that the Beta distribution is somewhat nice and symmetrical, if a bit obscure.

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More generally, this is why there's (still) a mild obsession with "conjugate" prior/likelihood distribution pairs - i.e. combinations of prior and likelihood that give analytically tractable posteriors[1] - despite the ability to get easily get results with MCMC.

[1] Yes, everyone wants a nice posterior. You're very clever.

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