基于概率论的分类方法：朴素贝叶斯

概述

朴素贝叶斯可以处理多类别问题，在数据较少的情况下仍然有效，但对于输入数据的准备方式较为敏感。

朴素贝叶斯是贝叶斯决策理论的一部分。

核心思想：选择高概率对应的类别，即选择具有最高概率的决策。

例如，当我们识别了一个障碍物，根据其特征，判断它是汽车的概率为60%，判定是卡车的概率为45%，判定是自行车的概率为30%，判定是行人的概率是5%，则最终我们会选择相信，该障碍物极有可能是一辆汽车，因为它是汽车的概率最大。

条件概率

设A、B是两个事件，且P(A) > 0，则称$P(B|A)=\frac{P(AB)}{p(A)}$为事件A发生的条件下事件B的条件概率
$$\{\begin{array}{ll}P(AB)=P(A)P(B|A),& \text{P(A) > 0}\\ P(AB)=P(B)P(A|B),& \text{P(B) > 0}\end{array}$$

贝叶斯公式

设 $A_1,A_2,...,A_n$ 是完备事件组，且$P(A_i)>0(i=1,2,...,n)$，B为任意事件，$P(B)>0$，则
$$P(A_k|B)=\frac{P(A_k)P(B|A_k)}{P(B)}=\frac{P(A_k)P(B|A_k)}{{\sum }_{i=1}^N(P(A_i)P(B|A_i))}$$

朴素贝叶斯分类器

2个假设

• 特征之间相互独立
• 每个特征同等重要

Example:文本分类

构建快速过滤器，屏蔽社区侮辱性言论。

• 首先从文本中构建词向量，将句子转化为向量

• 基于这些向量计算条件概率

• 构建分类器

创建一个名为 bayes.py的文件

from numpy import *

# 1. 词表到向量的转换函数
## 创建实验样本
def loadDataSet():
    postingList=[['my', 'dog', 'has', 'flea', 'problems', 'help', 'please'],
                 ['maybe', 'not', 'take', 'him', 'to', 'dog', 'park', 'stupid'],
                 ['my', 'dalmation', 'is', 'so', 'cute', 'I', 'love', 'him'],
                 ['stop', 'posting', 'stupid', 'worthless', 'garbage'],
                 ['mr', 'licks', 'ate', 'my', 'steak', 'how', 'to', 'stop', 'him'],
                 ['quit', 'buying', 'worthless', 'dog', 'food', 'stupid']]
    classVec = [0,1,0,1,0,1]    #1 is abusive, 0 not
    return postingList,classVec
 
 ## 创建一个包含在所有文档中出现的不重复词的列表
def createVocabList(dataSet):
    vocabSet = set([])  #create empty set
    for document in dataSet:
        vocabSet = vocabSet | set(document) #union of the two sets
    return list(vocabSet)

## 词汇表中的单词是否在输入文档中出现
def setOfWords2Vec(vocabList, inputSet):
    returnVec = [0]*len(vocabList)
    for word in inputSet:
        if word in vocabList:
            returnVec[vocabList.index(word)] = 1
        else: print "the word: %s is not in my Vocabulary!" % word
    return returnVec

# 2.朴素贝叶斯分类器训练函数
def trainNB0(trainMatrix,trainCategory):
    numTrainDocs = len(trainMatrix)
    numWords = len(trainMatrix[0])
    pAbusive = sum(trainCategory)/float(numTrainDocs)
    p0Num = ones(numWords); p1Num = ones(numWords)      #change to ones() 
    p0Denom = 2.0; p1Denom = 2.0                        #change to 2.0
    for i in range(numTrainDocs):
        if trainCategory[i] == 1:
            p1Num += trainMatrix[i]
            p1Denom += sum(trainMatrix[i])
        else:
            p0Num += trainMatrix[i]
            p0Denom += sum(trainMatrix[i])
    p1Vect = log(p1Num/p1Denom)          #change to log()
    p0Vect = log(p0Num/p0Denom)          #change to log()
    return p0Vect,p1Vect,pAbusive

# 3. 朴素贝叶斯分类函数
def classifyNB(vec2Classify, p0Vec, p1Vec, pClass1):
    p1 = sum(vec2Classify * p1Vec) + log(pClass1)    #element-wise mult
    p0 = sum(vec2Classify * p0Vec) + log(1.0 - pClass1)
    if p1 > p0:
        return 1
    else: 
        return 0
    
def bagOfWords2VecMN(vocabList, inputSet):
    returnVec = [0]*len(vocabList)
    for word in inputSet:
        if word in vocabList:
            returnVec[vocabList.index(word)] += 1
    return returnVec

def testingNB():
    listOPosts,listClasses = loadDataSet()
    myVocabList = createVocabList(listOPosts)
    trainMat=[]
    for postinDoc in listOPosts:
        trainMat.append(setOfWords2Vec(myVocabList, postinDoc))
    p0V,p1V,pAb = trainNB0(array(trainMat),array(listClasses))
    testEntry = ['love', 'my', 'dalmation']
    thisDoc = array(setOfWords2Vec(myVocabList, testEntry))
    print testEntry,'classified as: ',classifyNB(thisDoc,p0V,p1V,pAb)
    testEntry = ['stupid', 'garbage']
    thisDoc = array(setOfWords2Vec(myVocabList, testEntry))
    print testEntry,'classified as: ',classifyNB(thisDoc,p0V,p1V,pAb)

def textParse(bigString):    #input is big string, #output is word list
    import re
    listOfTokens = re.split(r'\W*', bigString)
    return [tok.lower() for tok in listOfTokens if len(tok) > 2] 

测试代码如下：

from bayes import *

def test_randy():
    print("******* 测试词表到向量的转换函数 createVocabList setOfWords2Vec ******* ")
    listOPosts, listClasses = loadDataSet()
    randyVocabList = createVocabList(listOPosts)
    print("============> randyVocabList \n", randyVocabList)
    print("============> cute existed? \n", setOfWords2Vec(randyVocabList, listOPosts[0]))
    print("============> garbage existed? \n", setOfWords2Vec(randyVocabList, listOPosts[3]))

    print("******* 测试朴素贝叶斯分类器训练函数 trainNB0 ******* ")
    trainMat = []
    for postinDoc in listOPosts:
        trainMat.append(setOfWords2Vec(randyVocabList, postinDoc))
    p0V, p1V, pAB = trainNB0(trainMat, listClasses)
    print("============> p0V \n", p0V)
    print("============> p1V \n", p1V)
    print("============> pAB \n", pAB)

    print("******* 测试朴素贝叶斯分类函数 testingNB ******* ")
    testingNB()

if __name__ == "main":
    test_randy

结果为：

******* 测试词表到向量的转换函数 createVocabList setOfWords2Vec ******* 
============> randyVocabList 
 ['maybe', 'so', 'not', 'stop', 'ate', 'cute', 'my', 'dog', 'to', 'park', 'worthless', 'how', 'posting', 'has', 'flea', 'help', 'him', 'love', 'buying', 'is', 'licks', 'food', 'garbage', 'problems', 'dalmation', 'I', 'steak', 'take', 'please', 'stupid', 'quit', 'mr']
============> cute existed? 
 [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0]
============> garbage existed? 
 [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0]
 
******* 测试朴素贝叶斯分类器训练函数 trainNB0 ******* 
============> p0V 
 [-3.25809654 -2.56494936 -3.25809654 -2.56494936 -2.56494936 -2.56494936
 -1.87180218 -2.56494936 -2.56494936 -3.25809654 -3.25809654 -2.56494936
 -3.25809654 -2.56494936 -2.56494936 -2.56494936 -2.15948425 -2.56494936
 -3.25809654 -2.56494936 -2.56494936 -3.25809654 -3.25809654 -2.56494936
 -2.56494936 -2.56494936 -2.56494936 -3.25809654 -2.56494936 -3.25809654
 -3.25809654 -2.56494936]
============> p1V 
 [-2.35137526 -3.04452244 -2.35137526 -2.35137526 -3.04452244 -3.04452244
 -3.04452244 -1.94591015 -2.35137526 -2.35137526 -1.94591015 -3.04452244
 -2.35137526 -3.04452244 -3.04452244 -3.04452244 -2.35137526 -3.04452244
 -2.35137526 -3.04452244 -3.04452244 -2.35137526 -2.35137526 -3.04452244
 -3.04452244 -3.04452244 -3.04452244 -2.35137526 -3.04452244 -1.65822808
 -2.35137526 -3.04452244]
============> pAB 
 0.5
 
******* 测试朴素贝叶斯分类函数 testingNB ******* 
['love', 'my', 'dalmation'] classified as:  0
['stupid', 'garbage'] classified as:  1
['love', 'my', 'dalmation'] classified as:  0
['stupid', 'garbage'] classified as:  1

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