 Solution

Let $Z$ be the unobserved part of the fourth count associated with the component of probability $\theta /3$. Its complementary subcount has probability $7/12$. Conditional on the observed fourth count and a parameter value,The complete-data log-likelihood now has the simple formThe constant term can depend on the completed counts, but not on $\theta$. The split therefore converts the troublesome $\log (7+4\theta )$ term into a binomial-like maximization problem. Since $Z$ is unobserved, it should be averaged conditionally in an EM algorithm, not guessed or treated as observed.