From the Mean-Square Error (MSE) formula, we have the relation:
- Variance: The inherent randomness or “spread” of the model’s output.
- K: The number of simulations you run.
To reduce the error, you can either:
- Increase K (Brute-Force): Run more simulations. This is effective but can be very expensive and time-consuming.
- Reduce Variance (Smart Method): Reformulate the problem to measure something with less inherent randomness. This is the goal of variance reduction techniques.
Analogy: The Dartboard
- High Variance: Trying to find the center of a large dartboard. Your throws are widely spread, so your average guess is noisy and less accurate.
- Low Variance: Aiming for a tiny target just around the bullseye. Your throws are tightly clustered. The same number of throws gives you a much more accurate average.
Variance reduction methods like Control Variates and MLMC are like replacing the big, noisy dartboard with a smaller, more focused target. They cleverly change the problem so that each simulation provides more precise information, leading to a more accurate result with less effort.