EyEular

如果一个变量在两个 clique 之间出现，那么这个变量一定在这两个 clique 的路径之间 learning is parameter estimation sufficient statistics $T(x)$ means x self or some transformation of x 先验概率可理解为统计概率，后验概率可理解为条件概率 In Bayesian probab... Continue reading...

Introduction of PGMWhy Using PGMFor a joint distribution $P(X_1,X_2,…,X_n)$, can be written in two ways All Dependent: $P(X_1,X_2,…,X_n) = P(X_1)P(X_2|X_1)P(X_2|X_1,X_2)…P(X_n|X_1,…,X_{n-1})$ pro:... Continue reading...

i.i.d sequence of random variables: too restrictive assumption completely dependent among random variables: hard to analysis balance between complete independence & complete dependence Classifi... Continue reading...

Bayesian InferenceBayesian Inference FrameworkWe aim to extract information about $\Theta$, based on observing a collection $X = (X_1,…,X_n)$ Unknow $\Theta$ treated as a random variable prior dis... Continue reading...

Conditional Expectation Given An EventIf $Y$ is a discrete r.v. E(Y|A) = \sum_y P(Y=y|A)If $Y$ is continuous r.v. E(Y|A) = \int_{-\infty}^{\infty} yf(y|A)dyApproximationImage a large number of $n$ ... Continue reading...

BasisDiscriminative & Generative Model 判别模型 Discriminative Model，又可以称为条件模型，或条件概率模型。估计的是条件概率分布(conditional distribution) 生成模型 Generative Model，又叫产生式模型。估计的是联合概率分布（joint probability distribution） ... Continue reading...

Beta-Binomial DistributionDefinition (Beta Distribution)An r.v. $X$ is said to have Beta distribution with parameters $a$ and $b$, if its PDF is f(x) = \frac{1}{\beta(a,b)}x^{a-1}(1-x)^{b-1},0 Continue reading...

Discrete Multivariate R.V.sDefinition (Joint CDF) The Joint CDf of r.v.s $X$ and $Y$ is the function $F_{X,Y}$ given by F_{X,Y}(x,y) = P(X\leq x,Y\leq y)Definition (Joint PMF) The Joint PMF of disc... Continue reading...

Eular 定理对于一个多面体 P，我们定义 v：定点数 e：棱边数 f：面数 Eular 定理：$v+f-e = 2$ 但是满足这个定理的多面体是有条件的： P 的任何两个顶点可以用一串棱相连接 反例：中空的立方体 P 上任意由直线段构成的圈，把 P 分割成两片 反例：螺帽柱状体 Eular 定理证明：我们首先来看看树形，这个在图论里面常常出现，树有个性质就是 $v-e=1... Continue reading...

Generating FunctionThree kinds of generating functions Probability Generating Function (PGF) : related to Z-transform Moment Generating Function (MGF) : related to Laplace transform Characteristic ... Continue reading...