Challenge: Convex Optimization

Problem statement

Suppose a new movie is released and you want to estimate the probability that a random viewer will like it. You assume that this probability follows a beta distribution with a fixed lower bound of $0$ and an upper bound of $1$. You also assume that the shape parameter of the beta distribution $\theta$ is unknown, and you want to find its maximum likelihood estimate.

Mathematical formulation

You collect some data by asking $n$ ...