kldiv {bayesmeta} R Documentation

## Kullback-Leibler divergence of two multivariate normal distributions.

### Description

Compute the Kullback-Leiber divergence or symmetrized KL-divergence based on means and covariances of two normal distributions.

### Usage

  kldiv(mu1, mu2, sigma1, sigma2, symmetrized=FALSE)


### Arguments

 mu1, mu2 the two mean vectors. sigma1, sigma2 the two covariance matrices. symmetrized logical; if TRUE, the symmetrized divergence will be returned.

### Details

The Kullback-Leibler divergence (or relative entropy) of two probability distributions p and q is defined as the integral

D_{\mathrm{KL}}(p\,||\,q) = \int_\Theta \log\Bigl(\frac{p(\theta)}{q(\theta)}\Bigr)\, p(\theta)\, \mathrm{d}\theta.

In the case of two normal distributions with mean and variance parameters given by (\mu_1, \Sigma_1) and (\mu_2, \Sigma_2), respectively, this results as

D_{\mathrm{KL}}\bigl(p(\theta|\mu_1,\Sigma_1)\,||\,p(\theta|\mu_2,\Sigma_2)\bigr) = \frac{1}{2}\biggl(\mathrm{tr}(\Sigma_2^{-1} \Sigma_1) + (\mu_1-\mu_2)^\prime \Sigma_2^{-1} (\mu_1-\mu_2) - d + \log\Bigl(\frac{\det(\Sigma_2)}{\det(\Sigma_1)}\Bigr)\biggr)

where d is the dimension.

The symmetrized divergence simply results as

D_{\mathrm{s}}(p\,||\,q)=D_{\mathrm{KL}}(p\,||\,q)+D_{\mathrm{KL}}(q\,||\,p).

### Value

The divergence (D_{\mathrm{KL}} \geq 0  or D_{\mathrm{s}} \geq 0 ).

### Author(s)

Christian Roever christian.roever@med.uni-goettingen.de

### References

S. Kullback. Information theory and statistics. John Wiley and Sons, New York, 1959.

C. Roever, T. Friede. Discrete approximation of a mixture distribution via restricted divergence. Journal of Computational and Graphical Statistics, 26(1):217-222, 2017. doi:10.1080/10618600.2016.1276840.

bmr.
kldiv(mu1=c(0,0), mu2=c(1,1), sigma1=diag(c(2,2)), sigma2=diag(c(3,3)))