Past exam of the options I course of the University of Oxford / 2013 / O / 2 / d

The further observation is independent of the earlier observations conditional on $\theta$. Integrate its conditional success probability against the posterior:The given factorial formula for the beta function gives this ratio as $A/(A+B)$. Thus the posterior predictive probability isIt is the posterior mean of the Bernoulli success parameter. Independence here is conditional: sharing the uncertain parameter generally makes past and future observations dependent before conditioning on it.