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 $\mathbf{V},\mathbf{W},\mathbf{B}$ in Monte Carlo Quadrature: Beyond Mean Estimation.

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

Mean

$$\mathbb{C}\text{ov}[{\mathbf{\Delta }}_{\alpha },{\mathbf{\Delta }}_{\beta }]=F_{\alpha \beta }\mathbb{C}\text{ov}[f_{\alpha },f_{\beta }]\in {\mathbb{R}}^{K\times K}$$

$$F_{\alpha \beta }=\frac{N_{{\alpha }^{\ast }\cap {\beta }^{\ast }}}{N_{{\alpha }^{\ast }}{N}_{{\beta }^{\ast }}}-\frac{N_{{\alpha }^{\ast }\cap \beta }}{N_{{\alpha }^{\ast }}{N}_{\beta }}-\frac{N_{\alpha \cap {\beta }^{\ast }}}{N_{\alpha }{N}_{{\beta }^{\ast }}}+\frac{N_{\alpha \cap \beta }}{N_{\alpha }{N}_{\beta }}$$

$$\mathbb{C}\text{ov}[{\mathbf{Q}}_0,{\mathbf{\Delta }}_{\alpha }]=G_{\alpha }\mathbb{C}\text{ov}[f_0,f_{\alpha }]\in {\mathbb{R}}^{K\times K}$$

$$G_{\alpha }=\frac{N_{0\cap {\alpha }^{\ast }}}{N_0{N}_{{\alpha }^{\ast }}}-\frac{N_{0\cap \alpha }}{N_0{N}_{\alpha }}$$

Variance

$$\mathbb{C}\text{ov}[{\mathbf{\Delta }}_{\alpha },{\mathbf{\Delta }}_{\beta }]=F_{\alpha \beta }{\mathbf{W}}_{\alpha \beta }+H_{\alpha \beta }{\mathbf{V}}_{\alpha \beta }\in {\mathbb{R}}^{K^2\times K^2}$$

$$\begin{array}{rl}{H}_{\alpha \beta }& =\frac{N_{{\alpha }^{\ast }\cap {\beta }^{\ast }}(N_{{\alpha }^{\ast }\cap {\beta }^{\ast }}-1)}{N_{{\alpha }^{\ast }}{N}_{{\beta }^{\ast }}(N_{{\alpha }^{\ast }}-1)(N_{{\beta }^{\ast }}-1)}-\frac{N_{{\alpha }^{\ast }\cap \beta }(N_{{\alpha }^{\ast }\cap \beta }-1)}{N_{{\alpha }^{\ast }}{N}_{\beta }(N_{{\alpha }^{\ast }}-1)(N_{\beta }-1)}\\ & -\frac{N_{\alpha \cap {\beta }^{\ast }}(N_{\alpha \cap {\beta }^{\ast }}-1)}{N_{\alpha }{N}_{{\beta }^{\ast }}(N_{\alpha }-1)(N_{{\beta }^{\ast }}-1)}+\frac{N_{\alpha \cap \beta }(N_{\alpha \cap \beta }-1)}{N_{\alpha }{N}_{\beta }(N_{\alpha }-1)(N_{\beta }-1)}\end{array}$$

$$\mathbb{C}\text{ov}[{\mathbf{Q}}_0,{\mathbf{\Delta }}_{\alpha }]=J_{\alpha }{\mathbf{V}}_{0\alpha }+G_{\alpha }{\mathbf{W}}_{0\alpha }\in {\mathbb{R}}^{K^2\times K^2}$$

$$J_{\alpha }=\frac{N_{0\cap {\alpha }^{\ast }}(N_{0\cap {\alpha }^{\ast }}-1)}{N_0{N}_{{\alpha }^{\ast }}(N_0-1)(N_{{\alpha }^{\ast }}-1)}-\frac{N_{0\cap \alpha }(N_{0\cap \alpha }-1)}{N_0{N}_{\alpha }(N_0-1)(N_{\alpha }-1)}$$

Mean and Variance

$$\begin{array}{r}\mathbb{C}\text{ov}[{\mathbf{\Delta }}_{\alpha },{\mathbf{\Delta }}_{\beta }]({\mathcal{Z}}_{\text{ACV}})=\left[\begin{array}{cc}\mathbb{C}\text{ov}[{\mathrm{\Delta }}_{\alpha }^{\mu },{\mathrm{\Delta }}_{\beta }^{\mu }]& \mathbb{C}\text{ov}[{\mathrm{\Delta }}_{\alpha }^{\mu },{\mathrm{\Delta }}_{\beta }^{\mathrm{\Sigma }}]\\ \mathbb{C}\text{ov}[{\mathrm{\Delta }}_{\beta }^{\mathrm{\Sigma }},{\mathrm{\Delta }}_{\alpha }^{\mu }]& \mathbb{C}\text{ov}[{\mathrm{\Delta }}_{\alpha }^{\mathrm{\Sigma }},{\mathrm{\Delta }}_{\beta }^{\mathrm{\Sigma }}]\end{array}\right]\in {\mathbb{R}}^{(K+K^2)\times (K+K^2)}\end{array}$$

$$\begin{array}{r}\mathbb{C}\text{ov}[{\mathbf{Q}}_0,{\mathbf{\Delta }}_{\alpha }]({\mathcal{Z}}_{\text{ACV}})=\left[\begin{array}{cc}\mathbb{C}\text{ov}[Q_0^{\mu },{\mathrm{\Delta }}_{\alpha }^{\mu }]& \mathbb{C}\text{ov}[Q_0^{\mu },{\mathrm{\Delta }}_{\alpha }^{\mathrm{\Sigma }}]\\ \mathbb{C}\text{ov}[Q_0^{\mathrm{\Sigma }},{\mathrm{\Delta }}_{\alpha }^{\mu }]& \mathbb{C}\text{ov}[Q_0^{\mathrm{\Sigma }},{\mathrm{\Delta }}_{\alpha }^{\mathrm{\Sigma }}]\end{array}\right]\in {\mathbb{R}}^{(K+K^2)\times (K+K^2)}.\end{array}$$

$$\mathbb{C}\text{ov}[{\mathrm{\Delta }}_{\alpha }^{\mu },{\mathrm{\Delta }}_{\beta }^{\mathrm{\Sigma }}]=F_{\alpha \beta }{\mathbf{B}}_{\alpha \beta }\in {\mathbb{R}}^{K\times (K+K^2)}$$

$$\mathbb{C}\text{ov}[Q_0^{\mu },{\mathrm{\Delta }}_{\alpha }^{\mathrm{\Sigma }}]=G_{\alpha }{\mathbf{B}}_{0\alpha }\in {\mathbb{R}}^{K\times (K+K^2)}$$