Control variates: Difference between revisions

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In [[Monte Carlo methods]], one or more '''control variates''' may be employed to achieve [[variance reduction]] by exploiting the [[correlation]] between statistics.
==Underlying Principle==
Let the [[Parameter#Statistics|parameter]] of interest be <math>\mu</math>, and assume we have a statistic <math>m</math> such that <math>\mathbb{E}\left[m\right]=\mu</math>. If we are able to find another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> and <math>\rho_{mt}=\textrm{corr}\left[m,t\right]</math> are known values, then
In the case that <math>\sigma_m</math>, <math>\sigma_t</math>, and/or <math>\rho_{mt}</math> are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain [[least squares]] system; therefore this technique is also known as '''regression sampling'''.