MLE课程笔记

Random Variables 随机变量

Discrete Random Variables 离散随机变量

1. Bernoulli distribution伯努利（二项）分布

    X = {0, 1} 
    pmf p(x) = Bern(x|q), where 
    p(1) = Bern(1|q) = Pr[x = 1] = q (we also write p(x = 1)) 
    p(0) = Bern(0|q) = Pr[x = 0] = 1 1 q (we also write p(x = 0))
   

2. Categorical distribution 类别分布

   X = {0, 1, ..., C-1}
   pmf x ∼ p(x) = Cat(x|q), where
   probability vector q = [q0, q1, ..., qCC 1] with
   qk = q0 + q1 + ... + qCC 1 = 1
   p(k) = Cat(k|q) = Pr[x = k] = qk (we also write p(x = k)) 
   

one-hot vector

Continuous Random Variables 连续随机变量

Gaussian variable 高斯变量

MATLAB 代码

	Description of any MATLAB function: help < namefunction >
	Generate a Bernoulli rv Bern(q): binornd(1, q) or (rand(1) < q)
	Generate N Bernoulli rvs Bern(q): binornd(1, q, N, 1) or (rand(N, 1) < q)
	Generate a categorical rv Cat(q): mnrnd(1, q)
	Generate N categorical rvs Cat(q): mnrnd(1, q, N) (produces one-hot vectors as rows)
	Generate a Gaussian rv N (m,s2): normrnd(m, s) or (m + s ∗ randn(1))
	Gaussian pdf N (x|m,s2): normpdf(x, m, s)
	Plot a function y = f (x): plot(x, y)

Expectation期望

1. Bernoulli rv 伯努利分布：

2. categorical rv 类别分布：
   

3. gaussion 高斯分布
   Ex=x^2 +σ^2

方差Variance

1. Bernoulli RV伯努利变量
   Var=q(1-q)

2. Gaussian RV 高斯变量
   Var(x)=σ^2

3. categorical rv 类别变量
   

Linear Algebra 线性代数

L * 1 vector

1. 一个x向量 L * 1 vector
   

2. transpose of vector x 转置矩阵
   

3. normalize a vector 正则化矩阵（求方向）
   

4. measures the cosine of the angle θ between vectors x and y 两个向量的夹角
   

L * M matrix 矩阵

1. a L * M matrix
   

2. transpose matrix 转置矩阵