Flat-topped Probability Density Functions for Mixture Models
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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(𝐱)∝[cosh([(𝐱-𝐦)^𝖳Σ^-1(𝐱-𝐦)]^n/2)+cosh(r^n)]^-1 where 𝐱,𝐦∈ℝ^n, Σ is an n× 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.
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