Computing the variance of a conditional expectation via non-nested Monte Carlo

Computing the variance of a conditional expectation has often been of importance in uncertainty quantification, which is usually done via nested Monte Carlo simulation. Sun et al. (2011) has introduced an unbiased nested Monte Carlo estimator, which they call $1\frac{1}{2}$-level simulation since the optimal inner-level sample size can be bounded above as the total computational budget increases. In this letter we construct several unbiased non-nested Monte Carlo estimators for the variance of a conditional expectation based on the so-called pick-freeze scheme due to Sobol' (1990), which was originally introduced for computing variance-based sensitivity indices.