Why does the video use rejection sampling to obtain samples from the zero-variance density $g^*$?

Conditions

• Direct sampling from the original distribution $X \sim \mathrm{Normal}(0,1)$ is possible.
• The acceptance rule is $X_i > \gamma$.

Reasoning, step by step

1. Draw a large batch of unconditioned standard-normal samples.
2. Retain only the samples satisfying $X_i > \gamma$.
3. Treat the retained samples as samples from $g^*$.

Example

The narrator explains that because $g^*$ is the distribution of $X$ conditioned on the rare event happening, one can sample many $X$'s and keep only those which exceed the threshold, explicitly naming this procedure rejection sampling.

Common misconceptions

• Believing that rejection sampling is used to sample from the original distribution.
• Thinking that the retained samples are drawn from a different distribution than $g^*$.

Watch the explanation

Explore next

• How are the Normal proposal parameters chosen from the retained samples?
• What is the zero-variance density $g^*$ in the improved cross-entropy method?