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The Conjugate Beta Prior

Hello. I'm having a problem with trying to figure out this proof that shows the beta distribution is conjugate to the binomial distribution (picture attached). I understand it until the third row, but I got confused with this step from the third to the fourth row. I would appreciate a simple explanation. Thank you in advance.

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To go from the third to fourth row just ignore factors that are constant with respect to $\pi$. That is, let

$$\binom{n}{y}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}=k$$

so

$$p(\pi|y)=k \cdot \pi^y (1-\pi)^{(n-y)}\pi^{(\alpha-1)}(1-\pi)^{(\beta-1)}$$

or

$$p(\pi|y) \propto \pi^y(1-\pi)^{(n-y)}\pi^{(\alpha-1)}(1-\pi)^{(\beta-1)}$$

The idea's to deal just with the kernel of the probability distribution, knowing you can always put the normalizing constant back in later.

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  • $\begingroup$ Does anyone have a link to a proof that does not rely on proportionality, but rather works out the normalization constants as well? $\endgroup$ – j.a.gartner Nov 5 '17 at 0:46

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