Thus the posterior predictive is the (compound) beta-binomial distribution: B b(x \mid M, \widehat{\alpha}, \widehat{\beta}) \triangleq\left(\begin{array}{c} M \\ x \end{array}\right) \frac{B(x+\widehat{\alpha}, M-x+\widehat{\beta})}{B(\widehat{\alpha}, \widehat{\...