I want to calculate the Kullbach Leibler Divergence of two multivariate Gaussians as in

KL divergence between two multivariate Gaussians.

At one point one has to solve the following expression (which is also equation 380 in the matrix cookbook):

$$ E_q[(x-\mu_1)^T\Sigma_2^{-1}(\mu_1-\mu_2)]$$

where the expectation is taken across another multivariate gaussian $$ q(x) = N(\mu_1, \Sigma_1)$$.

And this should evaluate to zero, but I dont understand how and why


Note that $ (x - \mu_1)\Sigma_2^{-1}(\mu_1 - \mu_2)$ is just a linear function of x. You can write it as $(x-b)A$ where $b = \mu_1$ and $A = \Sigma_2^{-1}(\mu_1 - \mu_2)$ are constants.

By linearity of expectation, $E[(x-b)A] = (E[x] - b)A$. Note that $E[x] = b =\mu_1$ and thus the whole expression is zero.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.