统计学名词介绍

介绍

本文主要介绍概率论经常出现的名词和缩略形式

Cumulative distribution function（CDF） 累计分布函数

$$F\_X(x) = P(X \le x) = \sum\_{x\_i \le x} f\_X(x\_i).$$

Probability distribution（PD）

Probability Mass Function(PMF) 概率密度函数

这个和下一个PDF类似，但是用在discrete parameter上。所有可能的组合为1

Probability density function（PDF） 概率密度函数

$$f\_X(x) = P(X = x).$$ 离散情况 函数$f\_X$称为PDF. ### 联合密度函数 $$f(x,y) = P(X=x,Y=y)$$

frequentist inference and Bayesian inference

mean squared error, or MSE

贝叶斯模型

首先是先验概率，假设model为$P[0.1, 0.2, \cdots, 0.9]$的概率值为多少多少.
然后得到Model的likehood函数：$P(data | Model)$
算后验概率，$P(Model | data)$，注意这个对每个model而言的和为1.

$$P(Model | data) = \frac{P(data | Model) \times P(Model)}{P(data)} = \frac{P(data | Model) \times P(Model)}{\sum P(data | Model) \times P(Model)}$$

从prior到poster就是贝叶斯推断。使用一开始的贝叶斯模型得到的poster概率来预测新的参看第五题

Union Distribution

他的PDF是flat的

Beta Function

$${ \mathrm {B} (x,y)={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}.}$$

Wiki

Beta distribution

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution.
贝叶斯的prior和poster与beta function关系：http://stats.stackexchange.com/questions/58564/help-me-understand-bayesian-prior-and-posterior-distributions
Wiki

Conjugacy

analogous(类似的)
根据观察改变poster

P-value

检验$H_0$假设是否正确。

Poison

$${\displaystyle {\frac {\lambda ^{k}e^{-\lambda }}{k!}}}$$

Gamma distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The common exponential distribution and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:

1. With a shape parameter k and a scale parameter θ.
2. With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
3. With a shape parameter k and a mean parameter μ = k/β. $${\frac {1}{\Gamma (k)\theta ^{k}}}x^{k\,-\,1}e^{-{\frac {x}{\theta }}}$$ $$Mean = \scriptstyle \mathbf {E} [X]=k\theta$$

$$k^{*} = k + \sum x\_i$$ $$\theta^{*} = \frac{\theta}{n \theta + 1}$$

confidence interval

$$L, U = pe \pm se^{*} cv$$

Credible Intervals

The differecen between confidence interval and credible interval

Bayersian -> decision theory

Posterior distribution = prior + data
Bayersian -> decision theory -> loss function

• linear decision, the loss function is L1 loss, and minimum point is median.
• square decision, the loss function is L2. And the optimal is mean
• 0/1, the loss function is L0.
  For the posterior distribution, we can get the decision loss function distribution.

For example, there is two competing hypothesis: $H_1$,$H_2$.

$$P(H\_1\ true \vert data) = posterior\ probability\ of\ H\_1$$

L(d) means decide and the loss of this decision.
Expected losses: $E(L(d)) = \sum P(H_i \vert data)L(d) $

协方差

用来表示两个随机变量关系的统计量。
方差可以写成：

$$var(X) = \frac{\sum\_{i}(X\_i-\overline{X})(X\_i-\overline{X})}{n-1}$$

类似，协方差：

$$cov(X,Y)=\frac{\sum\_{i}(X\_i-\overline{X})(Y\_i-\overline{Y})}{n-1}$$

结果含义就是正值，两者呈现正相关。负值则相反。
协方差矩阵就是在高维情况下产生的。n维数据就会有$\dbinom{n}{2}$个协方差

$$\Sigma \_{ij}=\mathrm {cov} (X\_{i},X\_{j})=\mathrm {E} {\begin{bmatrix}(X\_{i}-\mu \_{i})(X\_{j}-\mu \_{j})\end{bmatrix}}$$ $$\mu\_{i} = E(X\_{i})$$

统计学三大分布

多项分布

极大似然估计（MLE）

似然函数$L_n(\theta) = \Pi_{i=1}^n f(X_i;\theta)$.
极大似然估计记为$\hat{\theta}_n$, 是使似然函数最大$\theta$的值。

参考

[1] 协方差-WIkipedia： https://en.wikipedia.org/wiki/Covariance_matrix

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