  Determining the strength of non-linear statistical dependencies between two
variables is a crucial matter in many research fields. The established measure
for quantifying such relations is the mutual information. However, estimating
mutual information from limited samples is a challenging task. Since the mutual
information is the difference of two entropies, the existing Bayesian
estimators of entropy may be used to estimate information. This procedure,
however, is still biased in the severely under-sampled regime. Here we propose
an alternative estimator that is applicable to those cases in which the
marginal distribution of one of the two variables---the one with minimal
entropy---is well sampled. The other variable, as well as the joint and
conditional distributions, can be severely undersampled. We obtain an estimator
that presents very low bias, outperforming previous methods even when the
sampled data contain few coincidences. As with other Bayesian estimators, our
proposal focuses on the strength of the interaction between two discrete
variables, without seeking to model the specific way in which the variables are
related. A distinctive property of our method is that the main data statistics
determining the amount of mutual information is the inhomogeneity of the
conditional distribution of the low-entropy variable in those states (typically
few) in which the large-entropy variable registers coincidences.
