一. Probability
1. 条件概率与独立性
, with
Equivalently,
If A and B are independent,
If，则称B对A有利(favourable/ probability-increasing)
If，则称B对A不利(unfavourable/ probability-decreasing)
Favourability is symmetric, B is probability-increasing for A if and only if A is probability-increasing for B.
2. 加法法则与全概率法则
At least one event occurs
Law of total probability
3. 随机变量与分布
随机变量 X： Ω → ℝ 将结果映射到实数轴。离散随机变量具有概率质量函数（Probability Mass Function），连续随机变量具有概率密度函数（Probability Density Function），混合分布可同时包含两者。
4. 概率密度
For a continuous random variable,does not imply
.
The density can be viewed as
对于联合连续的(X, Y),
二. Bayes' Theorem
1. Discrete Case
For discrete random variables X and Y,
Using the law of total probability,
新信念 = 旧信念 × 新证据
- 你原本觉得 X 有多可能（先验 P (X)）
- 现在看到了证据 Y，它在不同 X 下出现的概率不同（似然 P (Y|X)）
- 两者相乘，就得到了看到 Y 之后你应该有多确信 X（后验 P (X|Y)）
举个例子：
2. Continuous Case
For continuous random variables X and Y,
三. Bayesian Statistical Inference
1. Basic setup
- Anobservation model
, which determines thelikelihood
.
- Aprior distribution
Together they define the join distribution
2.The Posterior Distribution
- The Parameter Distribution after Observing Data
Hence,
The posterior from one analysis becomes the prior for the next:
