常用结论的证明记录

高斯分布的微分熵

$X\sim \mathcal{N}(\mu ,{\sigma }^2)$，${\displaystyle f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}}$，其微分熵推导过程如下：

$$\begin{array}{rl}h(X)& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{-1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})\cdot \ln [\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})]dx\\ & =\frac{-1}{\sqrt{2\pi {\sigma }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})\cdot (\ln (2\pi {\sigma }^2{)}^{-1/2}-\frac{(x-\mu {)}^2}{2{\sigma }^2})dx\\ & =\frac{1}{2}\ln (2\pi {\sigma }^2){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})dx\\ & +{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{(x-\mu {)}^2}{2{\sigma }^2}\cdot \frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})dx\\ & =\frac{1}{2}\ln (2\pi {\sigma }^2)+\frac{1}{2{\sigma }^2}E(X-\mu {)}^2\\ & =\frac{1}{2}\ln (2\pi {\sigma }^2)+\frac{1}{2}=\frac{1}{2}\ln (2\pi e{\sigma }^2)\text{(nats)}\end{array}$$

又因为 $H_b(X)={\log }_{b}a\cdot H_a(X)$，于是取 $b=e,a=2$ $$⟹\frac{1\text{ nat}}{x\text{ bits}}=\ln 2⟹x=\frac{1}{\ln 2}=\frac{\log e}{\log 2}=\log e\text{ bits}$$ 所以 $$\frac{1}{2}\ln (2\pi e{\sigma }^2)\text{ nats}=\frac{1}{2}\cdot \frac{\log (2\pi e{\sigma }^2)}{\log e}\cdot \log e\text{ bits}=\frac{1}{2}\log (2\pi e{\sigma }^2)\text{ bits}$$

概率

Bounds on tail probabilities

Markov’s inequality: For any r.v. $X$ and constant $a>0$,

$$P(|X|\ge a)\le \frac{E|X|}{a}.$$

Let $Y=\frac{|X|}{a}$. We need to show that $P(Y\ge 1)\le E(Y)$. Note that

$$I(Y\ge 1)\le Y,$$ since if $I(Y\ge 1)=0$ then $Y\ge 0$, and if $I(Y\ge 1)=1$ then $Y\ge 1$ (because the indicator says so). Taking the expectation of both sides, we have Markov’s inequality.

Chebyshev’s inequality: Let $X$ have mean $\mu$ and variance ${\sigma }^2$. Then for any $a>0$,

$$P(|X-\mu |\ge a)\le \frac{{\sigma }^2}{a^2}.$$

By Markov’s inequality,

$$P(|X-\mu |\ge a)=P((X-\mu {)}^2\ge a^2)\le \frac{E(X-\mu {)}^2}{a^2}=\frac{{\sigma }^2}{a^2}.$$

Chernoff inequality: For any r.v. $X$ and constants $a>0$ and $t>0$,

$$P(X\ge a)\le \frac{E(e^{tX})}{e^{ta}}.$$

The transformation $g$ with $g(x)=e^{tx}$ is invertible and strictly increasing. So by Markov’s inequality, we have

$$P(X\ge a)=P(e^{tX}\ge e^{ta})\le \frac{E(e^{tX})}{e^{ta}}.$$

Law of large numbers

Assume we have i.i.d. $X_1,X_2,X_3,\dots$ with finite mean $\mu$ and finite variance ${\sigma }^2$. For all positive integers $n$, let

$${\overline{X}}_n=\frac{X_1+\cdots +X_n}{n}$$

be the sample mean of $X_1$ through $X_n$. The sample mean is itself an r.v., with mean $\mu$ and variance ${\sigma }^2/n$:

$$\begin{array}{rl}E({\overline{X}}_n)& =\frac{1}{n}E\left(\sum _{i=1}^n{X}_i\right)=\frac{1}{n}\sum _{i=1}^{n}E(X_i)=\mu ,\\ \text{Var}({\overline{X}}_n)& =\frac{1}{n^2}\text{Var}\left(\sum _{i=1}^n{X}_i\right)=\frac{1}{n^2}\sum _{i=1}^n\text{Var}(X_i)=\frac{{\sigma }^2}{n}.\end{array}$$

Strong law of large numbers The sample mean ${\overline{X}}_n$ converges to the true mean $\mu$ pointwise as $n\to \mathrm{\infty }$, with probability 1. In other words,

$$P(\underset{n\to \mathrm{\infty }}{lim}{\overline{X}}_n=\mu )=1,\text{ or }{\overline{X}}_n\stackrel{a.s.}{⟶}\mu .$$

Weak law of large numbers For all $ϵ>0$, $P(|{\overline{X}}_n-\mu |>ϵ)\to 0$ as $n\to \mathrm{\infty }$. (This is called convergence in probability.) In other words,

$$\underset{n\to \mathrm{\infty }}{lim}P({\overline{X}}_n=\mu )=1.$$

Fix $ϵ>0$, by Chebyshev’s inequality,

$$P(|{\overline{X}}_n-\mu |>ϵ)\le \frac{{\sigma }^2}{n{ϵ}^2}.$$

As $n\to \mathrm{\infty }$, the right-hand side goes to 0, ans so must the left-hand side.

References

1. Blitzstein, Joseph K, and Hwang, Jessica. “Introduction to probability.”