Flat-topped Probability Density Functions for Mixture Models

TL;DR

This paper introduces flat-topped probability density functions based on logistic functions, which are continuous, adaptable, and computationally tractable, suitable for mixture models to improve fit and simplicity.

Contribution

It proposes a new class of flat-topped PDFs with a specific mathematical form, enhancing mixture model flexibility and computational efficiency.

Findings

The proposed PDFs are continuous and nearly uniform around the mode.

They are adaptable to various distribution shapes from bell to rectangular.

The PDFs are computationally advantageous for parameter estimation.

Abstract

This paper investigates probability density functions (PDFs) that are continuous everywhere, nearly uniform around the mode of distribution, and adaptable to a variety of distribution shapes ranging from bell-shaped to rectangular. From the viewpoint of computational tractability, the PDF based on the Fermi-Dirac or logistic function is advantageous in estimating its shape parameters. The most appropriate PDF for $n$-variate distribution is of the form: $p\left(\mathbf{x}\right)\propto\left[\cosh\left(\left[\left(\mathbf{x}-\mathbf{m}\right)^{\mathsf{T}}\boldsymbol{\Sigma}^{-1}\left(\mathbf{x}-\mathbf{m}\right)\right]^{n/2}\right)+\cosh\left(r^{n}\right)\right]^{-1}$ where $\mathbf{x},\mathbf{m}\in\mathbb{R}^{n}$, $\boldsymbol{\Sigma}$ is an $n\times n$ positive definite matrix, and $r>0$ is a shape parameter. The flat-topped PDFs can be used as a component of mixture models in machine learning to improve goodness of fit and make a model as simple as possible.