Probabilistic Circuits for Cumulative Distribution Functions
This work provides a theoretical foundation for efficient inference in probabilistic circuits by extending representations to CDFs, which is incremental for researchers in probabilistic modeling and machine learning.
The paper tackles the problem of representing cumulative distribution functions (CDFs) using probabilistic circuits, showing that for binary, discrete, and continuous variables, CDF and probability mass/density function representations can be transformed into each other efficiently, with polynomial-time or leaf-modification methods.
A probabilistic circuit (PC) succinctly expresses a function that represents a multivariate probability distribution and, given sufficient structural properties of the circuit, supports efficient probabilistic inference. Typically a PC computes the probability mass (or density) function (PMF or PDF) of the distribution. We consider PCs instead computing the cumulative distribution function (CDF). We show that for distributions over binary random variables these representations (PMF and CDF) are essentially equivalent, in the sense that one can be transformed to the other in polynomial time. We then show how a similar equivalence holds for distributions over finite discrete variables using a modification of the standard encoding with binary variables that aligns with the CDF semantics. Finally we show that for continuous variables, smooth, decomposable PCs computing PDFs and CDFs can be efficiently transformed to each other by modifying only the leaves of the circuit.