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Mutual information quantifies the dependence between two random variables andremains invariant under diffeomorphisms. In this paper, we explore thepointwise mutual information profile, an extension of mutual information thatmaintains this invariance. We analytically describe the profiles ofmultivariate normal distributions and introduce the family of finedistributions, for which the profile can be accurately approximated using MonteCarlo methods. We then show how fine distributions can be used to study thelimitations of existing mutual information estimators, investigate the behaviorof neural critics used in variational estimators, and understand the effect ofexperimental outliers on mutual information estimation. Finally, we show howfine distributions can be used to obtain model-based Bayesian estimates ofmutual information, suitable for problems with available domain expertise inwhich uncertainty quantification is necessary.

The Mixtures and the Neural Critics: On the Pointwise Mutual Information Profiles of Fine Distributions