# is there a computationally efficient formula for computing the mutual information between two continuous variables?

I need to compute the mutual information between two continuous variables. Below is an equation shown to compute the mutual information between a variable $X$ and $Y$.

$I(X;Y) = \int_Y \int_X p(x,y) \log{ \left(\frac{p(x,y)}{p(x)\,p(y)} \right) } \; dx \,dy$

I found this paper at, http://odin.mdacc.tmc.edu/~pqiu/pdf/FastPairMI_revision.pdf. they compute the mutual information as follows.

$I(X;Y) = \frac{1}{M} \sum_i \log { \frac{M \sum_j e^{-\frac{1}{2h^2}((x_i-x_j)^2+(y_i-y_j)^2)} } {\sum_j e^{-\frac{1}{2h^2} (x_i-x_j)^2} \sum_j e^{-\frac{1}{2h^2} (y_i-y_j)^2} } }$

Yet, here is another example at http://www.imt.liu.se/people/magnus/cca/tutorial/node16.html. although not very clear, i think $C_{XX}$ is the covariance of $X$ against itself and $C_{YY}$ is the covariance of $Y$ against itself.

$I(X;Y) = \frac{1}{2} \log \left( \frac{|C_{XX}| |C_{YY}|}{|C|} \right)$

My question is: are there more efficient ways to compute the mutual information between two continuous variables?

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