Meaning
Importance sampling techniques assign numerical values to individual data points to adjust for discrepancies between a proposal distribution and a target distribution. Using likelihood weighting allows a researcher to estimate the properties of a rare event by simulating from a highly frequent one. Each sample is multiplied by a ratio that accounts for its probability under the two different models.
This method stops being effective when the distributions have no overlap or when weights become excessively large.
Weight Assignment
Mathematical ratios scale the influence of every simulated path to match the target reality. In likelihood weighting, points that are more probable in the target than the proposal receive a greater value. This process ensures that the final average is unbiased.
It requires precise knowledge of the density functions involved.
Bias Correction
Adjustments to the sample mean prevent the estimator from following the shape of the simulation instead of the target. Without likelihood weighting, the results would only describe the proposal distribution. This correction is vital when studying edge cases like extreme market crashes.
Success depends on selecting a proposal that covers the entire range of the target.
Variance Reduction
Strategic selection of the proposal density minimizes the error in the final estimate. If the proposal is well-chosen, the likelihood weighting method produces a result with reduced variance than simple random sampling. This efficiency reduces the computational energy required to reach a specific confidence level.
Poorly chosen proposals lead to weight instability and unreliable figures.
