We now compute the posterior distribution for \(\theta\): \[\pi(\theta|x\ns_1,\ldots,x\ns_N)={f(x\ns_1,\ldots,x\ns_N|\theta)\,\pi(\theta)\over\int_0^1\!d\theta'\, f(x\ns_1,\ldots,x\ns_N|\theta')\,\pi(...We now compute the posterior distribution for \(\theta\): \[\pi(\theta|x\ns_1,\ldots,x\ns_N)={f(x\ns_1,\ldots,x\ns_N|\theta)\,\pi(\theta)\over\int_0^1\!d\theta'\, f(x\ns_1,\ldots,x\ns_N|\theta')\,\pi(\theta')} ={\theta^{X+\alpha-1} (1-\theta)^{N-X+\beta-1}\over\SB(X+\alpha,N-X+\beta)}\ .\] Thus, we retain the form of the Beta distribution, but with updated parameters, \[\begin{split} \alpha'&=X+\alpha\\ \beta'&=N-X+\beta\ . \end{split}\] The fact that the functional form of the prior is retaine…
