Monte Carlo Method to Approximate Variance

The Monte Carlo estimate for the variance, $\mathbb{V}[Y]$, is given by the formula:

$$\mathbb{V}[Y]\approx \frac{1}{K-1}\sum _{i=1}^K(\mathcal{M}(x^{(i)})-{\tilde{\mu }}_K{)}^2$$

• Generate Output Realizations: A set of $K$ independent random input realizations, $\{x^{(i)}{\}}_{i=1}^K$, is generated. The model is then run for each of these inputs to get a corresponding set of output realizations, $\mathcal{M}(x^{(i)})$.

• Estimate the Mean: The mean of the output is estimated by averaging the output realizations from the previous step. This mean estimate is denoted as ${\tilde{\mu }}_K$.

• Calculate Sum of Squared Differences: For each output $\mathcal{M}(x^{(i)})$, the squared difference between it and the estimated mean ${\tilde{\mu }}_K$ is calculated. These squared differences are then summed up for all $K$ realizations.

• Normalize the Sum: The sum is multiplied by the factor $\frac{1}{K-1}$. The notes mention that this factor is used to create an “un-biased estimator,” although using $\frac{1}{K}$ is also possible.