连续正态分布随机变量的熵
《机器学习数学基础》第 416 页给出了连续型随机变量的熵的定义,并且在第 417 页以正态分布为例,给出了符合 \(N(0,\sigma^2)\) 的随机变量的熵。
注意:在第 4 次印刷以及之前的版本中,此处有误,具体请阅读勘误表说明
1. 推导(7.6.6)式
假设随机变量服从正态分布 \(X\sim N(\mu,\sigma^2)\) (《机器学习数学基础》中是以标准正态分布为例,即 \(X\sim N(0,\sigma^2)\) )。
根据《机器学习数学基础》的(7.6.1)式熵的定义:
\[
H(X)=-\int f(x)\log f(x)\text{d}x\tag{7.6.1}
\]
其中,\(f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\) ,是概率密度函数。根据均值的定义,(7.6.1)式可以写成:
\[
H(X)=-E[\log f(x)]
\]
将 \(f(x)\) 代入上式,可得:
\[\begin{split}
\begin{split}
H(X)&=-E\left[\log(\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}})\right]
\\&=-E\left[\log(\frac{1}{\sqrt{2\pi}\sigma})+\log(e^{-\frac{(x-\mu)^2}{2\sigma^2}})\right]
\\&=-E\left[\log(\frac{1}{\sqrt{2\pi}\sigma})\right]-E\left[\log(e^{-\frac{(x-\mu)^2}{2\sigma^2}})\right]
\\&=\frac{1}{2}\log(2\pi\sigma^2)-E\left[-\frac{1}{2\sigma^2}(x-\mu)^2\log e\right]
\\&=\frac{1}{2}\log(2\pi\sigma^2)+\frac{\log e}{2\sigma^2}E\left[(x-\mu)^2\right]
\\&=\frac{1}{2}\log(2\pi\sigma^2)+\frac{\log e}{2\sigma^2}\sigma^2\quad(\because E\left[(x-\mu)^2\right]=\sigma^2,参阅 332 页 (G2)式)
\\&=\frac{1}{2}\log(2\pi\sigma^2)+\frac{1}{2}\log e
\\&=\frac{1}{2}\log(2\pi e\sigma^2)
\end{split}
\end{split}\]
从而得到第 417 页(7.6.6)式。
2. 推导多维正态分布的熵
对于服从正态分布的多维随机变量,《机器学习数学基础》中也假设服从标准正态分布,即 \(\pmb{X}\sim N(0,\pmb{\Sigma})\) 。此处不失一般性,以 \(\pmb{X}\sim N(\mu,\pmb{\Sigma})\) 为例进行推导。
注意:《机器学习数学基础》第 417 页是以二维随机变量为例,书中明确指出:不妨假设 \(\pmb{X}=\begin{bmatrix}\pmb{X}_1\\\pmb{X}_2\end{bmatrix}\) ,因此使用的概率密度函数是第 345 页的(5.5.18)式。
下面的推导,则考虑 \(n\) 维随机变量,即使用 345 页(5.5.19)式的概率密度函数:
\[
f(\pmb{X})=\frac{1}{\sqrt{(2\pi)^n|\pmb{\Sigma}|}}\text{exp}\left(-\frac{1}{2}(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right)
\]
根据熵的定义(第 416 页(7.6.2)式)得:
\[\begin{split}
\begin{split}
H(\pmb{X})&=-\int f(\pmb{X})\log(f(\pmb{X}))\text{d}\pmb{x}
\\&=-E\left[\log N(\mu,\pmb{\Sigma})\right]
\\&=-E\left[\log\left((2\pi)^{-n/2}|\pmb{\Sigma}|^{-1/2}\text{exp}\left(-\frac{1}{2}(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right)\right)\right]
\\&=-E\left[-\frac{n}{2}\log(2\pi)-\frac{1}{2}\log(|\pmb{\Sigma}|)+\log\text{exp}\left(-\frac{1}{2}(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right)\right]
\\&=\frac{n}{2}\log(2\pi)+\frac{1}{2}\log(|\pmb{\Sigma}|)+\frac{\log e}{2}E\left[(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right]
\end{split}
\end{split}\]
下面单独推导:\(E\left[(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right]\) 的值:
\[\begin{split}
\begin{split}
E\left[(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right]&=E\left[\text{tr}\left((\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right)\right]
\\&=E\left[\text{tr}\left(\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})(\pmb{X}-\pmb{\mu})^{\text{T}}\right)\right]
\\&=\text{tr}\left(\pmb{\Sigma^{-1}}E\left[(\pmb{X}-\pmb{\mu})(\pmb{X}-\pmb{\mu})^{\text{T}}\right]\right)
\\&=\text{tr}(\pmb{\Sigma}^{-1}\pmb{\Sigma})
\\&=\text{tr}(\pmb{I}_n)
\\&=n
\end{split}
\end{split}\]
所以:
\[\begin{split}
\begin{split}
H(\pmb{X})&=\frac{n}{2}\log(2\pi)+\frac{1}{2}\log(|\pmb{\Sigma}|)+\frac{\log e}{2}E\left[(\pmb{X}-\pmb{\mu})^{\text{T}}\pmb{\Sigma}^{-1}(\pmb{X}-\pmb{\mu})\right]
\\&=\frac{n}{2}\log(2\pi)+\frac{1}{2}\log(|\pmb{\Sigma}|)+\frac{\log e}{2}n
\\&=\frac{n}{2}\left(\log(2\pi)+\log e\right)+\frac{1}{2}\log(|\pmb{\Sigma}|)
\\&=\frac{n}{2}\log(2\pi e)+\frac{1}{2}\log(|\pmb{\Sigma}|)
\end{split}
\end{split}\]
当 \(n=2\) 时,即得到《机器学习数学基础》第 417 页推导结果:
\[
H(\pmb{X})=\log(2\pi e)+\frac{1}{2}\log(|\pmb{\Sigma}|)=\frac{1}{2}\log\left((2\pi e)^2|\pmb{\Sigma|}\right)
\]
参考资料
[1]. Entropy of the Gaussian[DB/OL]. https://gregorygundersen.com/blog/2020/09/01/gaussian-entropy/ , 2023.6.4
[2]. Entropy and Mutual Information[DB/OL]. https://gtas.unican.es/files/docencia/TICC/apuntes/tema1bwp_0.pdf ,2023.6.4
[3]. Fan Cheng. CS258: Information Theory[DB/OL]. http://qiniu.swarma.org/course/document/lec-7-Differential-Entropy-Part1.pdf , 2023.6.4.
[4]. Keith Conrad. PROBABILITY DISTRIBUTIONS AND MAXIMUM ENTROPY[DB/OL]. https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf, 2023.6.4.