Continuous \(\Theta\), Discrete \(K\)
coin with bias \(\Theta\); prior \(f_{\Theta}(\cdot)\)
\[f_{\Theta\lvert K}=\dfrac{f_{\Theta}(\theta)\,p_{K\lvert\Theta}(k\lvert\theta)}{p_K(k)}\] \[p_K(k)=\int f_{\Theta}(\theta')\,p_{K\lvert\Theta}(k\lvert\theta')\,\mathrm{d}\theta'\]The Beta Distribution
\[\theta^k(1-\theta)^{n-k}\]MAP Estimate
LMS Estimate
Note:
$$\int_{0}^{1}\theta^\alpha(1-\theta)^\beta\mathrm{d}\theta=\dfrac{\alpha!\,\beta!}{(\alpha+\beta+1)!}\qquad\text{for}\quad\alpha,\beta\geq0$$