机器学习——贝叶斯定理

7.1 贝叶斯定理

任务：

给定如下的队伍信息

  • 队0获胜的概率prob_win_0 = 0.65

  • 队1获胜时队1是主场的概率prob_win_1_home_1 = 0.75

  • 队0获胜时队1是主场的概率prob_win_0_home_1 = 0.3

请计算队1在客场取胜的概率p(Y=1|X=0)，即prob_home_0_win_1，小数点保留三位。

预期结果

待完善代码

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## 变量默认值代码

## 计算队1获胜的概率prob_win_1
prob_win_1 = 0.35

## 计算队1取胜时队0是东道主的概率
prob_win_1_home_0 = 0.25

## 计算队0取胜时队0是东道主的概率
prob_win_0_home_0 = 0.7

## 计算队0是东道主的概率
prob_home_0 = prob_win_1_home_0*prob_win_1+prob_win_0_home_0*(1-prob_win_1)
    
## 队1在客场取胜的概率
prob_home_0_win_1 = prob_win_1_home_0*prob_win_1/prob_home_0
print(round(prob_home_0_win_1,3))
0.161

7.2 朴素贝叶斯

任务

给定经过分词处理后的正常邮件和垃圾邮件各25封，对邮件内容分词后形成了词库列表vocab，存储到vocab_list.txt中。

  • 请自定义函数createVec(vocab, mail_vec)，输入词库列表vocab，和单个分词后的邮件内容列表mail_text，输出参数为向量化文本列表mail_vec。该自定义函数的作用是，将mail_text向量化，维度为vocab中的各个单词。比如，mail_text中含有vocab中的单词b，mail_vec的维度b的取值即为1。
  • 给定测试邮件内容列表test_mail，使用自定义函数返回向量化文本列表test_mail_vec。

给定数据的格式如下：vocab = [‘am’, ‘hacker’, ‘machine’,…]

预期结果

待补全代码

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import pandas as pd
import numpy as np
from numpy import linalg as LA

# 读取数据
vocab = pd.read_csv('vocab_list.txt',header=None).values.tolist()
li = []
for i in vocab:
    li.append(i[0])
vocab =li

test_mail = ['hi','peter','jose','town','want','meet','keep','things','going','interesting','stuff','let','know','eugene']


def createVector(vocab, mail_text): 
    
    ## 获取vocab的长度
    len_vocab = len(vocab)
    
    ## 零值初始化向量，长度为len_vocab
    mail_vec = np.zeros(len_vocab)
    
    ## 定义一个用来存储匹配单词位置的变量
    matchIdex = []

    ## 遍历mail_text中的元素
    for value in mail_text:
        
        ## 判断value在vocab中的位置，将mail_vec对应位置上的元素值加1
        if value in vocab:
            
            ## 获取value在vaocab中的索引value_index
            value_index=vocab.index(value)
            ## mail_vec中value_index位置上的元素赋值为1
            mail_vec[value_index]=1

            ## 记录下匹配单词的idex 
            matchIdex.append(value_index)
        else:
            continue
    
    ## 返回结果
    return mail_vec,matchIdex
## 获取test_mail_vec
test_mail_vec, matchIdex = createVector(vocab, test_mail)
print('该邮件编码为1的位置的索引向量为')
print(matchIdex)
该邮件编码为1的位置的索引向量为
[309, 465, 353, 606, 635, 401, 359, 590, 286, 342, 569, 370, 363, 226]

7.3 朴素贝叶斯的参数估计

任务

从50封邮件中选取各20封正常邮件和垃圾邮件，经过分词处理后，组成邮件训练集列表train_data。已知训练集的标签列表为train_label，约垃圾邮件标记为1，正常邮件标记为0；已知词库列表为vocab：

  • 使用createVector函数将train_data中的记录向量化，生成train_matrix；
  • 自定义函数createNBClassifier(train_matrix, train_label)函数，计算判定所需要的p(xi|c)，以及p(c)；

给定数据的格式如下：

train_data = [[‘I’, ‘am’, ‘data’, ‘hacker’…], [‘data’, ‘mining’,’machine’],…]

train_label = [1,0,1,1,…]

vocab = [‘am’, ‘hacker’, ‘machine’,…]

预期结果

前序代码

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#7.2 createVector 函数列表推导式的调用
def createVector(vocab, mail_text): 
    
    ## 获取vocab的长度
    len_vocab = len(vocab)
    len_data = len(mail_text)
    
    ## 零值初始化向量，长度为len_vocab
    mail_vec = np.zeros((len_data,len_vocab))
    
    ## 定义一个用来存储匹配单词位置的变量
    matchIdex = [[]]
    ## 遍历mail_text中的元素
    for i in range(len(mail_text)):
        for value in mail_text[i]:
            
            ## 判断value在vocab中的位置，将mail_vec对应位置上的元素值加1
            if value in vocab:
                
                ## 获取value在vaocab中的索引value_index
                value_index=vocab.index(value)
                ## mail_vec中value_index位置上的元素赋值为1
                mail_vec[i][value_index]=1

                ## 记录下匹配单词的idex 
                #matchIdex[i].append(value_index)
            else:
                continue
    
    ## 返回结果
    return mail_vec

待补全代码

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import pandas as pd
import numpy as np
from numpy import linalg as LA
#数据处理与读取
vocab = pd.read_csv('vocab_list.txt', header=None).values.tolist()
#print(vocab)
li = []
for i in vocab:
    li.append(i[0])
vocab =li

train_data = pd.read_csv('train_data.txt', header=None).values.tolist()
#print(train_data)
li = []
for i in train_data:
    li.append(i[0].split(" "))
train_data  =li

train_label = pd.read_csv('train_label.txt', header=None).values.tolist()
li = []
for i in train_label:
    li.append(i[0])
train_label  =li


## 使用列表推导式将train_data向量化，结果保存在train_matrix
train_matrix = createVector(vocab, train_data)
## 分离掉 matchIdex


def createNBClassifier(train_matrix, train_label):
    import numpy as np
    ## 获取训练集的数量num
    num = len(train_data)
    '''
    train_label = train_label.values.tolist()
    li = []
    for i in train_label:
        li.append(i[0])
    train_label = li
    '''
    ## 获取train_matrix的维度num_col
    num_col = len(train_matrix)

    ## 统计c = 1下x_{i}的出现的频次列表，形如[num(x_1|c = 1), num(x_2|c=1),...];同理 c = 0
    ## 为了防止出现概率为零的情况，我们将所有词出现的频次初始化为1
    p1_num = np.ones(len(vocab))
    p0_num = np.ones(len(vocab))

    ## 垃圾邮件和正常邮件出现的单词的数量  
    p1_vec_denom = 0
    p0_vec_denom = 0

    for index in range(num):
        if train_label[index] == 1:
            ## 单词出现的位置值为1，不出现值为0，因此可以直接相加，累计词频，train_matrix的元素为list
            ## 需要转换为ndarray，便于向量直接相加
            ## 统计垃圾邮件中的单词数
            p1_num = p1_num + np.array(train_matrix[index])
            p1_vec_denom = p1_vec_denom + sum(train_matrix[index])
            
            ## 正常邮件
        else:
            ## 单词出现的位置值为1，不出现值为0，因此可以直接相加，累计词频，train_matrix的元素为list
            ## 需要转换为ndarray，便于向量直接相加
            ## 统计正常邮件中的单词数
            p0_num = p0_num + np.array(train_matrix[index])
            p0_vec_denom = p0_vec_denom + sum(train_matrix[index])

    ####根据从训练集统计的频次
    ##计算[p(x_i | c = 1), p(x_2|c = 1)...] [p(x_i) | c = 0), p(x_i|c=0),...]
    ####为了防止向下溢出，概率取对数;为了配合初始化值为1的频次，分母+2
    p1_vec = np.log(p1_num/(p1_vec_denom+2))
    p0_vec = np.log(p0_num/(p0_vec_denom+2))

    ## 计算垃圾邮件出现的概率
    pc1 = sum(train_label)/len(train_label)

    ## 计算正常邮件出现的概率
    pc0 = 1-pc1

    ##返回结果
    return p1_vec, p0_vec, pc1, pc0

## 使用自定义函数返回结果p1_vec, p0_vec, pc1, pc0
p1_vec, p0_vec, pc1, pc0=createNBClassifier(train_matrix, train_label)


print('pc1=',pc1)
print('pc0=',pc0)
print('p1_vec')
print(p1_vec)
print('p0_vec')
print(p0_vec)

pc1= 0.5
pc0= 0.5
p1_vec
[-4.70411013 -5.21493576 -4.70411013 -4.70411013 -4.11632347 -6.31354805
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 -4.92725369 -6.31354805 -6.31354805 -6.31354805 -6.31354805 -6.31354805
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 -6.31354805 -5.21493576 -6.31354805 -6.31354805 -6.31354805 -6.31354805
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 -6.31354805 -5.62040087 -6.31354805 -5.21493576 -6.31354805 -6.31354805
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 -4.52178858 -6.31354805 -6.31354805 -6.31354805 -6.31354805 -6.31354805
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7.4 贝叶斯估计

任务

将剩余的10封邮件内容向量化，组成测试集邮件内容列表test_matrix。给定训练集拟合生成的朴素贝叶斯模型返回预测测试集需要的概率p1_vec，p0_vec，pc1和pc0，以及测试集的真实标签列表test_label。

  • 自定义函数predict(test_data, p1_vec, p0_vec, pc1, pc0)函数，判断输入的邮件是否为垃圾邮件；
  • 使用自定义的预测函数，对test_data进行预测，输出预测标签列表pred_label；
  • 根据pred_label和test_label计算朴素贝叶斯模型的错误率error_rate。

给定的数据格式如下：

test_data = [[0,0,0,1,0,0,…], [1,0,0,0,1,0,…]]

test_label = [0,0,1,1,…]

p1_vec = np.array([-0.02, -0.003, …])

p0_vec = np.array([-0.02, -0.003, …])

pc1 = 0.5

pc0 = 0.5

预期结果![](

https://ai-studio-static-online.cdn.bcebos.com/9b876f11e7ce46b08352da01ebd518698363f0f132c44f7aa39430c9a43431d9)

前序依赖代码及数据

  • 用于向量化邮件的词典数据
  • 前序学习出的类条件分布概率 p1_vec，p0_vec，及类别分布先验 pc1，pc0
  • 用于向量化邮件的函数

前序代码操作步骤：

  • 运行7.2节中的 createVector()
  • 运行7.3节中的 p1_vec, p0_vec, pc1, pc0 = createNBClassifier(train_matrix, train_label) 获得类条件分布概率 p1_vec，p0_vec，及类别分布先验 pc1，pc0
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# 导入词典以便于将剩余10封邮件向量化
vocab = pd.read_csv('vocab_list.txt', header=None).values.tolist()
li = []
for i in vocab:
    li.append(i[0])
vocab =li


## 加载本次实验数据
test_label = pd.read_csv('test_label.txt', header=None).values.tolist()
li = []
for i in test_label:
    li.append(i[0])
test_label  =li

test_data = pd.read_csv('test_data.txt', header=None).values.tolist()
li = []
for i in test_data:
    li.append(i[0])
test_data = li
 

## 使用列表推导式将 test_data 向量化，结果保存在 test_matrix
test_matrix = createVector(vocab, test_data)
## 分离掉 matchIdex


def predict(test_matrix, p1_vec, p0_vec, pc1, pc0):
    import numpy as np
   
    ## 声明预测标签列表，初始值为0
    pred_label = np.zeros(len(test_label))
    
    ## 遍历测试机中的每一个样本
    for index, record in enumerate(test_matrix):
        ## 将list转换为ndarray
        record_array = np.array(record)
        
        ## 根据公式 log(p(c|x))正比于 log(p(x_1|c)p(x_2|c)...p(x_k|c)p(c))
        ## 计算测试样本属于垃圾邮件的概率
        p1 = sum(record_array * p1_vec) + np.log(pc1)
        
        ## 计算测试样本属于正常邮件的概率
        p0 = sum(record_array * p0_vec) + np.log(pc0)
        ## 确认预测标签
        if p1 > p0:
            ##将索引为index的测试标签值改为1
            pred_label[index] = 1
        else:
            continue
    
    ## 返回结果
    return pred_label

## 使用自定义的函数预测测试集的标签
pred_label =  predict(test_matrix, p1_vec, p0_vec, pc1, pc0)
## 获取预测标签与真实标签的差，diff类型为ndarray
#diff = [1*(pred_label[i]!=test_label[i]) for i in range(len(pred_label))]
diff=np.array(abs(pred_label-test_label))

## 计算错误率
error_rate = sum(diff)/len(diff)
## 打印错误率
print('预测错误率为')
print(round(error_rate, 2))
print('真实标签为')
print(test_label)
print('预测标签为')
print(pred_label)
print('预测标签是否不一致向量')
print(diff)

预测错误率为
0.4
真实标签为
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
预测标签为
[1. 1. 1. 1. 0. 1. 1. 1. 1. 1.]
预测标签是否不一致向量
[1. 1. 1. 1. 0. 0. 0. 0. 0. 0.]
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    缺失模块。
    1、请确保node版本大于6.2
    2、在博客根目录（注意不是yilia根目录）执行以下命令：
    npm i hexo-generator-json-content --save

    3、在根目录_config.yml里添加配置： 

      jsonContent:
    meta: false
    pages: false
    posts:
      title: true
      date: true
      path: true
      text: false
      raw: false
      content: false
      slug: false
      updated: false
      comments: false
      link: false
      permalink: false
      excerpt: false
      categories: false
      tags: true

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