Exercise 3.1 - MLE for the Beroulli/binomial model

We begin with (3.11), which is the likelihood function of a collection of the outcomes in a coin-toss experiment $𝒟$ w.r.t. the parameter $𝜃$, the probability of heads:

where $N_0$ and $N_1$ are the number of tails/heads respectively.

To decompose the differential into term-independent forms, taking logarithm:

Setting its derivative to zero:

yields (3.22):

where $N$ is the size of $𝒟$.

Of course one need not turn to the logarithmic field. Differentiating $p(𝒟|𝜃)$ w.r.t. $𝜃$ directly gives the same result. But taking a logarithm almost always simplifies the form and the deduction procedure.