Reminder

One has access to $n$ samples $x_1, \dots, x_n$ drawn from an unknown data distribution $p_{\mathrm{data}}$:

The goal is to generate new samples drawn from $p_{\mathrm{data}}$.

Maximum Log-Likelihood Estimation (MLE)

The main idea of maximum likelihood is the following:

• one restricts the search for a model to a family of distributions $p(\cdot | \theta)$ parameterized by $\theta$.
• for each parameter $\theta$, one evaluates the density $p( \cdot | \theta)$ on the data points $x_1, \dots, x_n$: $p(x_1, \dots, x_n | \theta)$
• then one chose the parameter $\theta$ that maximizes the likelihood of the observed data $p(x_1, \dots, x_n | \theta)$.

Examples