# Control Variates

Control variates offer a method to reduce the variance of Monte Carlo approximations under certain situations. Suppose we want to estimate $$\mu = \mathbb{E}[\varphi(x)] = \int \varphi(x)\pi(x) \: dx$$ and we have some unbiased estimator $$\hat{\mu}(\mathcal{X}) = \frac{1}{N}\sum_{x_s \in \mathcal{X}} f(x_s)$$, i.e. $$\mathbb{E}[\hat{\mu}]=\mu$$ and $$x_n \sim \pi(x)$$, with $$\mathcal{X} = \{x_1,\cdots,x_N\}$$.

The estimator $$\hat{\mu}^*(\mathcal{X}) = \hat{\mu}(\mathcal{X}) + c(b(\mathcal{X}) - \mathcal{E}[b(\mathcal{X})])$$ is called a control variate, and $$b$$ is called a baseline. Note that $$\mathbb{E}[\hat{\mu}^*] = \mu$$ as well, but can have lower variance if the baseline $$b$$ is correlated with the original estimator $$\hat{\mu}$$. To see this, apply the variance operator to $$\hat{\mu}^*$$: $\mathbb{V}[\hat{\mu}^*(\mathcal{X})] = \mathbb{V}[\hat{\mu}(\mathcal{X})] + c^2\mathbb{V}[\hat{\mu}^*(\mathcal{X})] + 2c\text{Cov}[\hat{\mu}(\mathcal{X}),\hat{\mu}^*(\mathcal{X})]$.

As a function of $$c$$, the right hand side is minimized when $$c = -\frac{\text{Cov}[\hat{\mu}(\mathcal{X}),\hat{\mu}^*(\mathcal{X})]}{\mathbb{V}[\hat{\mu}^*(\mathcal{X})]}$$, giving the variance of the control variate as $$\mathbb{V}[\hat{\mu}^*(\mathcal{X})] = (1 - \rho_{\hat{\mu}, b}^2)\mathbb{V}[\hat{\mu}(\mathcal{X})]$$ where $$\rho_{\hat{\mu},\rho}$$ is the correlation between $$\hat{\mu}$$ and $$b$$. Thus theres is a negative correlation between this correlation and the variance of the control variate.

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Created: 2022-03-13 Sun 21:45

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