一. Probability1. 条件概率与独立性, withEquivalently,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 occursLaw of total probability3. 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 Theorem1. Discrete CaseFor 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 CaseFor continuous random variables X and Y,三. Bayesian Statistical Inference1. Basic setupAnobservation model, which determines thelikelihood.Aprior distributionTogether they define the join distribution2.The Posterior Distribution- The Parameter Distribution after Observing DataHence,The posterior from one analysis becomes the prior for the next:四. Prior and Posterior Prediction1. Prior predictive distribution2. Posterior predictive distribution3. Beta-Binomial Prediction