function that returns the input that maximizes the expression.

## finding argmax

### direct optimization

Typical maximization system. Take derivative, set it to 0, solve, plug in, solve. THis is pretty bad during times are not differentiable.

### gradient ascent

We take steps following the direction

\begin{equation} \theta_{1j} = \theta_{0j} + \eta \pdv{LL(\theta_{0})}{\theta_{0j}} \end{equation}

## additional information

### argmax of log

see argmax of log