Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 206 / 1 / e / Solution

Let $I_0=\{i:y_i=0\}$ and $I_{+}=\{i:y_i>0\}$. Under the proposed zero-inflated negative binomial model, independence gives the likelihood functionThe positive-count factor uses the ordinary, untruncated negative binomial distribution. With $f_i=f_{\mathrm{NB}}(y_i;r,{\lambda }_i)$, the log-likelihood isFor latent indicators $a_i$, the complete-data log-likelihood iswhere $a_i=1$ is allowed only when $y_i=0$. This separation is what makes the expectation-maximization algorithm convenient.