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.
Best Answer
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.