Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
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Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

Note that, to answer the question, the correlation returned by maximal coupling is a function of the four parameters of the Gaussians, while the maximal correlation is always equal to one.

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

Note that, to answer the question, the correlation returned by maximal coupling is a function of the four parameters of the Gaussians, while the maximal correlation is always equal to one.

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source | link

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

The correlation associated with a maximal coupling can be derived from \begin{align*}\mathbb{E}[XY]&=\mathbb{E}[XY\mathbb{I}_{X=Y}]+\mathbb{E}[XY\mathbb{I}_{X\ne Y}]\\ &=\int x^2 p(x)\wedge q(x)\text{d}x+ \int\int xy \{p(x)-p(x)\wedge q(x)\}\{q(y)-p(y)\wedge q(y)\}\text{d}x\text{d}y \end{align*}

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Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Pierre Jacob on Statisfaction explains how to implement a maximal coupling between two distributions $p$ and $q$. Which is essentially a sequence of two accept-reject steps. From there you can try to estimate the correlation $\varrho$ between both Gaussian variates.

enter image description here

Here is for instance a comparison of empirical estimates of the correlations, depending on the values of $\mu$ and $\sigma$ (with the other Gaussian being standard):

enter image description here

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