Past exam of the statistics course of the University of Oxford / 2017 / 3 / a

Let $s={\sum }_i{x}_i$. The Bernoulli distribution likelihood function is ${\theta }^s(1-\theta {)}^{n-s}$. Multiplication by the beta distribution prior density and Bayes' theorem giveRecognizing the normalized beta distribution,This is Beta-binomial conjugacy; the same update applies whether the data are recorded as the individual Bernoulli trials or their success count.

Conditional on $\theta$, the future Bernoulli random variable is independent of the observed statistical sample and has success probability $\theta$. Its posterior predictive probability is therefore the posterior mean of $\theta$. Put $A=a+s$ and $B=b+n-s$. Thenusing the beta-gamma identity and the Gamma function recurrence. HenceThe whole posterior predictive distribution of $X_{n+1}$ is the Bernoulli distribution with this success probability.