Bivariate Kullback-Leibler Divergence

Estimates the directed Kullback-Leibler divergence between two bivariate continuous distributions by numerical integration of kernel density estimates. Also computes pairwise divergences among groups and exact divergences for discrete, bivariate normal, bivariate Pareto type II, and independent bivariate Weibull models. The kernel estimator follows Chackochan, Sankaran and Nair (2026) .

BivKLD

BivKLD estimates directed Kullback-Leibler (KL) divergence between bivariate distributions. Its kernel estimator follows Chackochan, Sankaran, and Unnikrishnan Nair (2026) and uses bivariate Gaussian kernel densities from the ks package.

Installation

Install the submitted source archive with:

install.packages("BivKLD_0.1.0.tar.gz", repos = NULL, type = "source")

Two-sample estimate

library(BivKLD)

set.seed(2026)
x <- cbind(rnorm(100), rnorm(100))
y <- cbind(rnorm(100, 0.5), rnorm(100, -0.25))

biv_kld(x, y, bandwidth = "normal")
biv_kld(y, x, bandwidth = "normal")

KL divergence is directed, so the two results need not be equal. The default bandwidth selector is unconstrained smoothed cross-validation ("scv"), as in the reference paper. The normal-scale selector above is useful for quick exploration.

Pairwise group comparisons

measurements <- iris[, c("Sepal.Length", "Sepal.Width")]
biv_kld_matrix(measurements, iris$Species, bandwidth = "normal")

Rows are source distributions and columns are comparison distributions, so entry [i, j] estimates D(group_i || group_j).

Exact calculations

biv_kld_discrete(c(0.2, 0.3, 0.1, 0.4),
                 c(0.1, 0.4, 0.2, 0.3))

biv_kld_normal(c(-2, 2), matrix(c(1, 1, 1, 2), 2),
               c(-2, 2), matrix(c(3, 3, 3, 6), 2))

Reference

Chackochan, R., Sankaran, P. G., & Unnikrishnan Nair, N. (2026). Bivariate Kullback-Leibler divergence. Communications in Statistics - Theory and Methods, 55(1), 292-312. doi:10.1080/03610926.2025.2496687