On the Normalizing Constant of the Continuous Categorical Distribution

@article{GordonRodrguez2022OnTN,
  title={On the Normalizing Constant of the Continuous Categorical Distribution},
  author={Elliott Gordon-Rodr{\'i}guez and Gabriel Loaiza-Ganem and Andres Potapczynski and John P. Cunningham},
  journal={ArXiv},
  year={2022},
  volume={abs/2204.13290}
}
Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical . This family enjoys remarkable mathematical simplicity; its density function resembles that of the Dirichlet distribution, but with a normalizing constant that can be written in closed form using elementary functions only. In spite of this mathematical simplicity, our… 

Figures from this paper

On a novel probability distribution for zero-laden compositional data

This paper reviews the key properties of the novel distribution and presents an application where it can be used for dimensionality reduction of compositional data, and highlights some underexplored connections between the machine learning and Compositional data analysis.

Data Augmentation for Compositional Data: Advancing Predictive Models of the Microbiome

This work extends the success of data augmentation to compositional data, i.e., simplex-valued data, which is of particular interest in the context of the human microbiome, and sets a new state-of-the-art for key disease prediction tasks including colorectal cancer, type 2 diabetes, and Crohn's disease.

References

SHOWING 1-10 OF 17 REFERENCES

Uses and Abuses of the Cross-Entropy Loss: Case Studies in Modern Deep Learning

This work identifies the potential for outperformance in categorical cross-entropy loss models, thereby highlighting the importance of a proper probabilistic treatment, as well as illustrating some of the failure modes thereof.

The continuous categorical: a novel simplex-valued exponential family

This work introduces a novel exponential family of distributions for modeling simplex-valued data - the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli.

The continuous Bernoulli: fixing a pervasive error in variational autoencoders

A new [0,1]-supported, single parameter distribution is introduced: the continuous Bernoulli, which patches this pervasive bug in VAE and suggests a broader class of performant VAE.

Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models

This compendium covers energy-based models, variational autoencoders, generative adversarial networks, autoregressive models, normalizing flows, in addition to numerous hybrid approaches, drawn under a single cohesive framework.

An Improved Method for Numerical Inversion of Laplace Transforms

An improved procedure for numerical inversion of Laplace transforms is proposed based on accelerating the convergence of the Fourier series obtained from the inversion integral using the trapezoidal

Numerical Methods for Laplace Transform Inversion

Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations. When solving such problems, in many cases it is fairly easy to obtain the

The Accurate Numerical Inversion of Laplace Transforms

Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour. The number n of points to be used is one of several parameters, in most cases yielding

What Every Computer Scientist Should Know About Floating-Point Arithmetic

The OPEN LOOKand Sun™ Graphical User Interfaces were developed by Sun Microsystems, Inc. for its users and licensees and Sun acknowledges the pioneering efforts of Xerox in researching and developing the concept of visual or graphical user interfaces for the computer industry.

mpmath: a Python library for arbitrary-precision floating-point arithmetic

The following example computes 50 digits of pi by numerically evaluating the Gaussian integral with mpmath.

Algorithm 368: Numerical inversion of Laplace transforms [D5]