Exercise 3.13 - Posterior predictive distribution for a batch of data with the dirichlet-multinomial model

The likelihood for Dirichlet-multinomial model is:

following the symbols defined in the textbook. The conjugate prior is the Dirichlet distribution:

where $𝜃$ is a $K$-dimension simplex. The (3.37) in the textbook mistake $𝜃$ for $\mathbf{\text{𝐱}}$.

The posterior distribution is another Dirichlet distribution with update:

To predict a new batch of data $\widetilde{𝒟}$, we begin with one sample $x\in \widetilde{𝒟}$:

where the expectation is computed w.r.t. the posterior Dirichlet distribution, hence is:

Finally,