先驗概率：

        在缺少某個前提下的變數概率，在機器學習中就是沒有訓練樣本，在訓練之前的初始概率：P（w）

後驗概率：

        在有了樣本資料以後，對變數進行概率的修正，得到的概率就是後驗概率，，例如g是樣本，則後驗概率是：P（w | g）

貝葉斯公式：

        從形式上講，貝葉斯公式通過先驗概率和似然函式求取後驗概率。

         P（w | g）= P(w) P(g | w)   /   P(g)

R 語言貝葉斯公式計算例子：

        先驗概率： 機器的狀態有兩種，工作working（概率是：0.9），或者損壞broken（概率是：0.1）

        似然概率： 在兩種狀態下，結果有好壞兩種， good or broken

good	broken
working	0.95	0.05
broken	0.7	0.3

         然後給出一組結果，"g", "b", "g", "g", "g", "g", "g", "g", "g", "b", "g", "b"， 求後驗概率

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)

這裡的全概率P(g) = P(g | w)P(w) + P(g | b)P(b)

下面是R程式碼

########################################################
# Illustration of function bayes to illustrate
# sequential learning in Bayes' rule
########################################################

bayes <- function(prior, likelihood, data){
  probs <- matrix(0, length(data) + 1, length(prior))
  dimnames(probs)[[1]] <- c("prior", data)
  dimnames(probs)[[2]] <- names(prior)
  probs[1, ] <- prior
  for(j in 1:length(data))
    probs[j+1, ] <- probs[j, ] * likelihood[, data[j]] /
    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 <- c(working = .9, broken = .1)

# outcomes are good (g) or broken (b)
# likelihood matrix gives probs of each outcome for each model

like.working <- c(g=.95, b=.05)
like.broken <- c(g=.7, b=.3)
likelihood <- rbind(like.working, like.broken)

# sequence of data outcomes

data <- c("g", "b", "g", "g", "g", "g", "g", "g", "g", "b", "g", "b")

# function bayes will computed the posteriors, one datum at a time
# inputs are the prior vector, likelihood matrix, and vector of data

posterior <- bayes(prior, likelihood, data)
posterior
 

執行結果：

        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