A Type Theory for Probability Density Functions

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There has been great interest in creating probabilistic programming languages to simplify the coding of statistical tasks; however, there still does not exist a formal language that simultaneously provides continuous probability distributions, the ability to naturally express custom probabilistic models, and Probability Density Functions (PDFs). This collection of features is necessary for mechanizing fundamental statistical techniques. The authors formalize the first probabilistic language that exhibits these features, and it serves as a foundational framework for extending the ideas to more general languages. Particularly novel are their type system for Absolutely Continuous (AC) distributions (those which permit PDFs) and their PDF calculation procedure, which calculates PDFs for a large class of AC distributions.