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Let $x$ and $y$ be two multivariate Gaussian variables, $x\sim\mathcal{N}(\mu_x,\Sigma_x)$, $y\sim\mathcal{N}(\mu_y,\Sigma_y)$. What is the value of the expectation.

$$ \operatorname{E}\left(e^{x^\top y}\right) $$

More generally, $x$ and $y$ might be correlated. In which case

$$ \begin{bmatrix} x \\ y \end{bmatrix} \sim\mathcal{N}\left( \begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}, \begin{bmatrix} \Sigma_{xx} & \Sigma_{xy} \\ \Sigma_{xy}^\top & \Sigma_{yy} \end{bmatrix} \right) $$

At first glance, this seems similar to the expectation $\operatorname{E}\left(e^{x}\right)$, which can be evaluated by completing the square in a Gaussian integral. However, $\operatorname{E}\left(e^{x^\top y}\right)$ has quadratic terms. Does anyone know if this has a closed-form solution, and if so what a good reference might be?

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  • $\begingroup$ You could probably simplify this by first looks at the univariate, independent case. $\endgroup$
    – John
    Oct 6 '17 at 19:58
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    $\begingroup$ As you hint, and as @John suggests, it's the obvious generalization of the univariate calculation. See en.wikipedia.org/wiki/… $\endgroup$
    – whuber
    Oct 6 '17 at 20:54
  • $\begingroup$ Quick note: the univariate case reveals that there are scenarios in which this expectation diverges. If $x$ is a standard Gaussian then $E(e^{x^2}) = \int \exp(x^2 - x^2 / 2) dx / \sqrt{2\pi} = \int \exp(x^2 / 2) dx / \sqrt{2\pi}$, which has negative variance and does not converge. $\endgroup$
    – MRule
    Oct 7 '17 at 11:16
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We can use the fact that $$E\left( e^{x^\top y} \right) = E_x\left( E_y\left( e^{x^\top y} \mid x \right) \right)$$ The conditional distribution of $y$ given $x$ is $$p(y \mid x) = \mathcal{N}\left( \mu_y + \Sigma_{xy}\Sigma_{yy}^{-1} \mu_x, \Sigma_{yy} - \Sigma_{xy}^\top \Sigma_{xx}^{-1} \Sigma_{xy} \right)$$ and so for fixed $x$ $$x^\top y \sim \mathcal{N}( \underbrace{x^\top \overbrace{\left( \mu_y + \Sigma_{xy}\Sigma_{yy}^{-1} \mu_x \right)}^u}_{\mu(x)}, \underbrace{x^\top \overbrace{\left( \Sigma_{yy} - \Sigma_{xy}^\top \Sigma_{xx}^{-1} \Sigma_{xy} \right)}^{A} x}_{\sigma^2(x)} ).$$ For fixed $x$, $e^{x^\top y}$ is therefore log-normally distributed and we have $$E_y\left(e^{x^\top y} \mid x \right) = e^{\mu(x) + \sigma^2(x) / 2}.$$ By completing the square and using the known normalization constant of the Gaussian distribution, it is possible to derive a closed-form solution. Perhaps somebody else can provide a more elegant or compact solution than my brute-force attempt: \begin{align} &\, E_x\left( \exp \left( \mu(x) + \sigma^2(x) / 2 \right) \right) \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp\left( -\frac{1}{2} (x - \mu_x)^\top \Sigma_{xx}^{-1} (x - \mu_x) + x^\top u + \frac{1}{2} x^\top A x \right) \, dx \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp\left( -\frac{1}{2} x^\top \Sigma_{xx}^{-1} x + x^\top \Sigma_{xx}^{-1} \mu_x - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x + x^\top u + \frac{1}{2} x^\top A x \right) \, dx \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp\left( -\frac{1}{2} x^\top (\Sigma_{xx}^{-1} - A) x + x^\top (\Sigma_{xx}^{-1} \mu_x + u) - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x \right) \, dx \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp( -\frac{1}{2} x^\top \underbrace{(\Sigma_{xx}^{-1} - A)}_{B} x + x^\top \underbrace{(\Sigma_{xx}^{-1} \mu_x + u)}_{Bv} - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x ) \, dx \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp\left( -\frac{1}{2} x^\top B x + x^\top B v - \frac{1}{2} v^\top B v + \frac{1}{2} v^\top B v - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x \right) \, dx \\ =& \frac{1}{|2\pi\Sigma_{xx}|^{1/2}} \int \exp\left( -\frac{1}{2} (x - v)^\top B (x - v) + \frac{1}{2} v^\top B v - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x \right) \, dx \\ =& \frac{|2\pi B^{-1}|^{1/2}}{|2\pi\Sigma_{xx}|^{1/2}} \exp\left( \frac{1}{2} v^\top B v - \frac{1}{2} \mu_x^\top \Sigma_{xx}^{-1} \mu_x \right) \end{align}

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  • $\begingroup$ If you would first diagonalize $\Sigma$, then in one step you could directly apply the univariate result. That would be simpler and much more revealing. $\endgroup$
    – whuber
    Oct 7 '17 at 14:43

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