What is the zero-variance density $g^*$ in the improved cross-entropy method for estimating $P(X>\gamma)$ ?

Q: What is the zero-variance density $g^*$ in the improved cross-entropy method for estimating $P(X>\gamma)$?

The zero-variance density $g^*$ is the conditional distribution of the random variable $X$ given that the rare event $X>\gamma$ occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

The zero-variance density

g^*

is the conditional distribution of the random variable

X

given that the rare event

X>\gamma

occurs. It is the ideal importance-sampling proposal that would make the estimator have zero variance.

Conditions

The target quantity is the rare-event probability $P(X>\gamma)$ .
$g^*$ is supported only on the region $x>\gamma$ .

Reasoning, step by step

Identify the rare event $X>\gamma$ .
Condition the original distribution of $X$ on this event.
Define $g^*(x)$ as the resulting conditional density.

Example

The notebook defines $g^*(x) = P(X = x | X > \gamma)$ , and the narrator calls it the zero-variance density, explaining it is the distribution of $X$ conditioned on the rare event happening.

Common misconceptions

Assuming $g^*$ is the unconditional distribution of $X$ .
Thinking $g^*$ has support over the entire real line rather than just $x>\gamma$ .

Watch the explanation

YouTubeImproved cross-entropy and original cross-entropy methods

0:39 – 0:58Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.

What is the zero-variance density g∗g^*g∗ in the improved cross-entropy method for estimating P(X>γ)P(X>\gamma)P(X>γ)?