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Distributions are fundamental statistical elements that play essentialtheoretical and practical roles. The article discusses experiences of trainingneural networks to classify univariate empirical distributions and to representthem on the two-dimensional latent space forcing disentanglement based on theinputs of cumulative distribution functions (CDF). The latent spacerepresentation has been performed using an unsupervised beta variationalautoencoder (beta-VAE). It separates distributions of different shapes whileoverlapping similar ones and empirically realises relationships betweendistributions that are known theoretically. The synthetic experiment ofgenerated univariate continuous and discrete (Bernoulli) distributions withvarying sample sizes and parameters has been performed to support the study.The representation on the latent two-dimensional coordinate system can be seenas an additional metadata of the real-world data that disentangles importantdistribution characteristics, such as shape of the CDF, classificationprobabilities of underlying theoretical distributions and their parameters,information entropy, and skewness. Entropy changes, providing an "arrow oftime", determine dynamic trajectories along representations of distributions onthe latent space. In addition, post beta-VAE unsupervised segmentation of thelatent space based on weight-of-evidence (WOE) of posterior versus standardisotopic two-dimensional normal densities has been applied detecting thepresence of assignable causes that distinguish exceptional CDF inputs.

AI Giving Back to Statistics? Discovery of the Coordinate System of Univariate Distributions by Beta Variational Autoencoder