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The iteration threshold $\gamma_i$ is defined as the empirical $(1-p)$-quantile of the current batch of samples drawn from the proposal distribution. The loop stops when this threshold $\gamma_i$ is greater than or equal to the target rare-event threshold $\gamma$.

Conditions: Samples are drawn from the current proposal distribution.; The elite fraction parameter $p$ is set to 0.05.; The target threshold is $\gamma=5$.

The iteration threshold $\gamma_i$ is defined as the empirical $(1-p)$-quantile of the current batch of samples drawn from the proposal distribution. The loop stops when this threshold $\gamma_i$ is greater than or equal to the target rare-event threshold $\gamma$.

Conditions: Samples are drawn from the current proposal distribution.; The elite fraction parameter $p$ is set to 0.05.; The target threshold is $\gamma=5$.

The elite samples are weighted by likelihood ratios because they are drawn from the current proposal distribution, not the original target distribution. The likelihood ratio corrects for this discrepancy, ensuring that the updated proposal parameters accurately reflect the maximum likelihood estimates under the original distribution.

Conditions: Elite samples are those with values above the iteration threshold $\gamma_i$.; The likelihood ratios are computed as the original density divided by the proposal density.

The elite samples are weighted by likelihood ratios because they are drawn from the current proposal distribution, not the original target distribution. The likelihood ratio corrects for this discrepancy, ensuring that the updated proposal parameters accurately reflect the maximum likelihood estimates under the original distribution.

Conditions: Elite samples are those with values above the iteration threshold $\gamma_i$.; The likelihood ratios are computed as the original density divided by the proposal density.

The video states that the earlier example 'cheated' because the rare event was not very rare, making direct rejection sampling from $g^*$ computationally feasible. In general, for truly rare events, direct rejection sampling would take too long, which motivates the use of the adaptive cross-entropy method.

Conditions: The context is comparing the improved cross-entropy method to the original cross-entropy method.; The rare event in the earlier example had a relatively high probability.

The video states that the earlier example 'cheated' because the rare event was not very rare, making direct rejection sampling from $g^*$ computationally feasible. In general, for truly rare events, direct rejection sampling would take too long, which motivates the use of the adaptive cross-entropy method.

Conditions: The context is comparing the improved cross-entropy method to the original cross-entropy method.; The rare event in the earlier example had a relatively high probability.