先验概率：

在缺少某个前提下的变量概率，在机器学习中就是没有训练样本，在训练之前的初始概率：P(w)

后验概率：

在有了样本数据以后，对变量进行概率的修正，得到的概率就是后验概率，，例如g是样本，则后验概率是：P(w | g)

贝叶斯公式：

从形式上讲，贝叶斯公式通过先验概率和似然函数求取后验概率。

P(w | g)= P(w) P(g | w)   /   P(g)

R 语言贝叶斯公式计算例子：

先验概率： 机器的状态有两种，工作working(概率是：0.9)，或者损坏broken(概率是：0.1)

似然概率： 在两种状态下，结果有好坏两种， good or broken

good

broken

working

0.95

0.05

broken

0.7

0.3

然后给出一组结果，"g","b","g","g","g","g","g","g","g","b","g","b"， 求后验概率

即 P(w | g), P(w | b), P(b | g), P(b | b)

例如， P(w | g)= P(w) P(g | w) / P(g)

这里的全概率P(g) = P(g | w)P(w) + P(g | b)P(b)

下面是R代码

########################################################

# Illustration of function bayes to illustrate

# sequential learning in Bayes' rule

########################################################

bayes

probs

dimnames(probs)[[1]]

dimnames(probs)[[2]]

probs[1, ]

for(j in 1:length(data))

probs[j+1, ]

sum(probs[j, ] * likelihood[, data[j]])

dimnames(probs)[[1]]

paste(0:length(data), dimnames(probs)[[1]])

data.frame(probs)

}

# quality control example

# machine is either working or broken with prior probs .9 and .1

prior

# outcomes are good (g) or broken (b)

# likelihood matrix gives probs of each outcome for each model

like.working

like.broken

likelihood

# sequence of data outcomes

data

# function bayes will computed the posteriors, one datum at a time

# inputs are the prior vector, likelihood matrix, and vector of data

posterior

posterior

执行结果：

working broken

0 prior 0.9000 0.10000

1 g 0.9243 0.07568

2 b 0.6706 0.32941

3 g 0.7342 0.26576

4 g 0.7894 0.21055

5 g 0.8358 0.16424

6 g 0.8735 0.12649

7 g 0.9036 0.09641

8 g 0.9271 0.07289

9 g 0.9452 0.05476

10 b 0.7421 0.25793

11 g 0.7961 0.20389

12 b 0.3942 0.60578