What is the final importance-sampling estimator for $P(X>\gamma)$ using the fitted Normal proposal?

Conditions

• Samples are drawn from the fitted Normal proposal, not the original distribution.
• The event indicator is $\mathbf{1}_{\{Z_j>\gamma\}}$.

Reasoning, step by step

1. Draw samples from the fitted Normal proposal.
2. Compute the likelihood ratio $f(Z_j)/q(Z_j)$ for each sample.
3. Multiply the likelihood ratio by the indicator $\mathbf{1}_{\{Z_j>\gamma\}}$.
4. Average the weighted indicators to obtain the estimate.

Example

The code computes `lrs = stats.norm.pdf(normals, 0, 1) / stats.norm.pdf(normals, mu, sigma)`, then `ests = lrs * (normals > gamma)`, then `lHat = ests.mean()`.

Common misconceptions

• Drawing samples from the original standard normal distribution instead of the proposal.
• Forgetting to multiply the likelihood ratio by the rare-event indicator.

Watch the explanation

Explore next

• How is the reported confidence interval constructed from the importance-sampling estimates?
• How are the Normal importance-sampling proposal parameters chosen from the retained $g^*$ samples?