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In many applications in biology, engineering and economics, identifyingsimilarities and differences between distributions of data from complexprocesses requires comparing finite categorical samples of discrete counts.Statistical divergences quantify the difference between two distributions.However, their estimation is very difficult and empirical methods often fail,especially when the samples are small. We develop a Bayesian estimator of theKullback-Leibler divergence between two probability distributions that makesuse of a mixture of Dirichlet priors on the distributions being compared. Westudy the properties of the estimator on two examples: probabilities drawn fromDirichlet distributions, and random strings of letters drawn from Markovchains. We extend the approach to the squared Hellinger divergence. Bothestimators outperform other estimation techniques, with better results for datawith a large number of categories and for higher values of divergences.

Bayesian estimation of the Kullback-Leibler divergence for categorical sytems using mixtures of Dirichlet priors