Delta-Based Covariance Formulas For Approximate Control Variates

The following lists the formula needed to use ACV to compute vector-valued statistics comprised of means, variances or both. The formulas use the expressions for \(\mat{V}, \mat{W}, \mat{B}\) in Monte Carlo Quadrature: Beyond Mean Estimation.

These covariancesdepend on how samples are allocated to the sets \(\rvset_\alpha,\rvset_\alpha^*\), which we call the sample allocation \(\mathcal{A}\). Specifically, \(\mathcal{A}\) specifies: the number of samples in the sets \(\rvset_\alpha,\rvset_\alpha^*, \forall \alpha\), denoted by \(N_\alpha\) and \(N_{\alpha^*}\), respectively; the number of samples in the intersections of pairs of sets, that is \(N_{\alpha\cap\beta} =|\rvset_\alpha \cap \rvset_\beta|\), \(N_{\alpha^*\cap\beta} =|\rvset_\alpha^* \cap \rvset_\beta|\), \(N_{\alpha^*\cap\beta^*} =|\rvset_\alpha^* \cap \rvset_\beta^*|\); and the number of samples in the union of pairs of sets \(N_{\alpha\cup\beta} = |\rvset_\alpha\cup \rvset_\beta|\) and similarly \(N_{\alpha^*\cup\beta}\), \(N_{\alpha^*\cup\beta^*}\).

Mean

\[\covar{\mat{\Delta}_\alpha}{\mat{\Delta}_\beta} = F_{\alpha\beta}\covar{f_\alpha}{f_\beta}\in\reals^{K\times K}\]

\[F_{\alpha\beta} = \frac{N_{\alpha^*\cap \beta^*}}{N_{\alpha^*}N_{\beta^*}} - \frac{N_{\alpha^*\cap \beta}}{N_{\alpha^*}N_{\beta}} - \frac{N_{\alpha\cap \beta^*}}{N_{\alpha}N_{\beta^*}} + \frac{N_{\alpha\cap \beta}}{N_{\alpha}N_{\beta}}\]

\[\covar{\mat{Q}_0}{\mat{\Delta}_\alpha} = G_{\alpha}\covar{f_0}{f_\alpha}\in\reals^{K\times K}\]

\[G_{\alpha} = \frac{N_{0\cap \alpha^*}}{N_{0}N_{\alpha^*}} - \frac{N_{0\cap \alpha}}{N_{0}N_{\alpha}}\]

Variance

\[\covar{\mat{\Delta}_\alpha}{\mat{\Delta}_\beta} = F_{\alpha\beta}\mat{W}_{\alpha\beta}+H_{\alpha\beta}\mat{V}_{\alpha\beta}\in\reals^{K^2\times K^2}\]

\[\begin{split}H_{\alpha\beta} &= \frac{N_{\alpha^*\cap \beta^*}(N_{\alpha^*\cap \beta^*}-1)}{N_{\alpha^*}N_{\beta^*}(N_{\alpha^*}-1)(N_{\beta^*}-1)} - \frac{N_{\alpha^*\cap \beta}(N_{\alpha^*\cap \beta}-1)}{N_{\alpha^*}N_{\beta}(N_{\alpha^*}-1)(N_{\beta}-1)} \\& \qquad\quad- \frac{N_{\alpha\cap \beta^*}(N_{\alpha\cap \beta^*}-1)}{N_{\alpha}N_{\beta^*}(N_{\alpha}-1)(N_{\beta^*}-1)} + \frac{N_{\alpha\cap \beta}(N_{\alpha\cap \beta}-1)}{N_{\alpha}N_{\beta}(N_{\alpha}-1)(N_{\beta}-1)}\end{split}\]

\[\covar{\mat{Q}_0}{\mat{\Delta}_\alpha} = J_\alpha\mat{V}_{0\alpha}+G_{\alpha}\mat{W}_{0\alpha}\in\reals^{K^2\times K^2}\]

\[J_{\alpha} = \frac{N_{0\cap \alpha^*}(N_{0\cap \alpha^*}-1)}{N_{0}N_{\alpha^*}(N_{0}-1)(N_{\alpha^*}-1)} - \frac{N_{0\cap \alpha}(N_{0\cap \alpha}-1)}{N_{0}N_{\alpha}(N_{0}-1)(N_{\alpha}-1)}\]

Mean and Variance

\[\begin{split}\covar{\mat{\Delta}_\alpha}{\mat{\Delta}_\beta}(\rvset_\text{ACV}) = \begin{bmatrix}\covar{\Delta^\mu_\alpha}{\Delta^\mu_\beta} & \covar{\Delta^{\mu}_\alpha}{\Delta^{\Sigma}_\beta}\\\covar{\Delta^{\Sigma}_\beta}{\Delta^{\mu}_\alpha} & \covar{\Delta^{\Sigma}_\alpha}{\Delta^{\Sigma}_\beta}\end{bmatrix}\in\reals^{(K+K^2)\times (K+K^2)}\end{split}\]

\[\begin{split}\covar{\mat{Q}_0}{\mat{\Delta}_\alpha}(\rvset_\text{ACV}) = \begin{bmatrix}\covar{Q^\mu_0}{\Delta^\mu_\alpha} & \covar{Q^{\mu}_0}{\Delta^{\Sigma}_\alpha} \\ \covar{Q^{\Sigma}_0}{\Delta^{\mu}_\alpha} & \covar{Q^{\Sigma}_0}{\Delta^{\Sigma}_\alpha}\end{bmatrix} \in\reals^{(K+K^2)\times (K+K^2)}.\end{split}\]

\[\covar{\Delta^{\mu}_\alpha}{\Delta^{\Sigma}_\beta} = F_{\alpha\beta}\mat{B}_{\alpha\beta}\in\reals^{K\times (K+K^2)}\]

\[\covar{Q^{\mu}_0}{\Delta^{\Sigma}_\alpha}=G_\alpha\mat{B}_{0\alpha}\in\reals^{K\times (K+K^2)}\]