论文标题
混合模型的平面概率密度函数
Flat-topped Probability Density Functions for Mixture Models
论文作者
论文摘要
本文研究了各处连续的概率密度函数（PDF），几乎围绕分布方式均匀，并且适应从钟形到矩形的各种分布形状。从计算障碍的角度来看，基于费米 - dirac或logistic函数的PDF在估计其形状参数方面是有利的。最合适的PDF用于$ n $ variate分发的形式： $ p \ left（\ mathbf {x} \ right）\ propto \ left [\ cosh \ left（\ left [\ left（\ Mathbf {x}} - \ MathBf {m Mathbf {m} \ right）^{\ Mathsf {\ mathsf {t}}}} \ bo ldsymbolς^{ - 1} \ left（\ mathbf {x} - \ mathbf {m} \ right）\ right] \ right]^{n/2} \ right）+\ cosh \ cosh \ left（r^{n} \ right）^{ - 1} $其中$ \ mathbf {x}，\ mathbf {m} \ in \ mathbb {r}^{n} $，$ \boldsymbolς$是$ n \ times n $ nos stautial distriite matrix，$ r> 0 $是形状参数。扁平的PDF可以用作机器学习中混合模型的组成部分，以提高拟合度并使模型尽可能简单。
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: $p\left(\mathbf{x}\right)\propto\left[\cosh\left(\left[\left(\mathbf{x}-\mathbf{m}\right)^{\mathsf{T}}\boldsymbolΣ^{-1}\left(\mathbf{x}-\mathbf{m}\right)\right]^{n/2}\right)+\cosh\left(r^{n}\right)\right]^{-1}$ where $\mathbf{x},\mathbf{m}\in\mathbb{R}^{n}$, $\boldsymbolΣ$ 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.
