1.贝叶斯公式

• 条件概率
  
  p(B|A)=p(AB)p(A)
  
  则
  p(AB)=p(A)p(B|A)
• 全概率公式
  
  p(A)=p(B1)p(A|B1)+p(B2)p(A|B2)+...+p(Bn)p(A|Bn)
• 贝叶斯公式
  
  p(Bi|A)=p(ABi)p(A)=p(A|Bi)p(Bi)Σj=0np(A|Bj)p(Bj)
  
  该公式给出了在事件A下，事件Bi发生的概率的计算方法。通常，将此公式成为后验概率公式，即在已知观察量A后得出的参数B的分布。其中p(Bi)称为先验概率，是人们根据经验给出的参数Bi的分布。
  贝叶斯方法与最大似然法的区别就在于引入了先验概率，通过先验概率可以避免最大似然法所带来的过拟合问题。

2.朴素贝叶斯方法

• 对于B={B1,B2...Bn}，其条件概率可表示为
  
  p(B|A)=p(B1|A)p(B2|A,B1)p(B3|A,B1,B2)...p(Bn|A,B1,...,Bn?1)
  然而在实际情况中，等式右边的公式很难计算出来。故我们做出一个较强的假设，即Bi是相互独立的，这样条件概率可以表示为
  
  p(B|A)=p(B1|A)p(B2|A)...p(Bn|A)
  这就是朴素贝叶斯方法。当然在实际情况中，这种相互独立的假设往往是不成立的，然而其还是可以在一定程度上给出对数据的描述。
• 根据这个假设，我们可以分别计算p(Bi|A)∝p(A|Bi)p(Bi)若对?j≠i, 有p(Bi|A)>p(Bj|A)则A就可归为Bi类

3.实例

在训练过程中，需要计算两个概率：
* 先验概率p(Bi)=Num(Bi)Num(B)
* 条件概率p(A|Bi)=Num(A,Bi)Num(Bi)

from numpy import *

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]
    return postingList, classVec

def createVocabList(dataSet):
    vocabSet = set([])
    for document in dataSet:
        vocabSet = vocabSet | set(document)
    return list(vocabSet)

def setOfWord2Vec(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

def trainNB0(trainMatrix, trainCategory):
    numTrainDocs = len(trainMatrix)
    numWords = len(trainMatrix[0])
    pAbusive = sum(trainCategory)/float(numTrainDocs)
    p0Num = ones(numWords)
    p1Num = ones(numWords)
    p0Denom = 2.0; p1Denom = 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)
    p0Vect =log(p0Num/p0Denom)
    return p0Vect, p1Vect, pAbusive

def classifyNB(vec2Classify, p0Vec, p1Vec, pClass1):
    p1 = sum(vec2Classify*p1Vec) + log(pClass1)
    p0 = sum(vec2Classify*p0Vec) + log(1.0-pClass1)
    if p1 > p0:
        return 1
    else:
        return 0


def testingNB():
    listOPosts, listClasses = loadDataSet()
    myVocabList = createVocabList(listOPosts)
    trainMat = []
    for postinDoc in listOPosts:
        trainMat.append(setOfWord2Vec(myVocabList, postinDoc))
    p0V, p1V, pAb = trainNB0(array(trainMat), array(listClasses))
    testEntry = ['love', 'my', 'dalmation']
    thisDoc = array(setOfWord2Vec(myVocabList, testEntry))
    print testEntry, 'classified as:', classifyNB(thisDoc, p0V, p1V, pAb)
    testEntry=['stupid', 'garbage']
    thisDoc = array(setOfWord2Vec(myVocabList, testEntry))
    print testEntry, 'classified as:', classifyNB(thisDoc, p0V, p1V, pAb)

def bagOfWords2VecMN(vocabList, inputSet):
    returnVec = [0]*len(vocabList)
    for word in inputSet:
        if word in vocabList:
            returnVec[vocabList.index(word)]+=1
    return returnVec


def textParse(bigString):
    import re
    listOfTokens = re.split(r'\W*', bigString)
    return [tok.lower() for tok in listOfTokens if len(tok)>2]

def spamTest():
    docList = []; classList=[]; fullText=[]
    for i in range(1, 26):
        wordList = textParse(open('email/spam/%d.txt' % i).read())
        docList.append(wordList)
        fullText.extend(wordList)
        classList.append(1)
        wordList = textParse(open('email/ham/%d.txt' % i).read())
        docList.append(wordList)
        fullText.extend(wordList)
        classList.append(0)
    vocabList = createVocabList(docList)
    trainingSet = range(50);
    testSet = []
    for i in range(10):
        randIndex = int(random.uniform(0, len(trainingSet)))
        testSet.append(trainingSet[randIndex])
        del(trainingSet[randIndex])
    trainMat=[]; trainClasses=[]
    for docIndex in trainingSet:
        trainMat.append(setOfWord2Vec(vocabList, docList[docIndex]))
        trainClasses.append(classList[docIndex])
    p0V, p1V, pSpam=trainNB0(array(trainMat), array(trainClasses))
    errorCount = 0
    for docIndex in testSet:
        wordVector = setOfWord2Vec(vocabList, docList[docIndex])
        if classifyNB(array(wordVector), p0V, p1V, pSpam) != classList[docIndex]:
            errorCount += 1
    print 'the error rate is: ',float(errorCount)/len(testSet)


def calcMostFreq(vocabList, fullText):
    import operator
    freqDict={}
    for token in vocabList:
        freqDict[token] = fullText.count(token)
    sortedFreq = sorted(freqDict.iteritems(), key=operator.itemgetter(1), reverse=True)
    return sortedFreq[:30]

def localWords(feed1, feed0):
    import feedparser
    docList=[]; classList=[]; fullText=[]
    minLen = min(len(feed1['entries']), len(feed0['entries']))
    for i in range(minLen):
        wordList = textParse(feed1['entries'][i]['summary'])
        docList.append(wordList)
        fullText.extend(wordList)
        classList.append(1)
        wordList = textParse(feed0['entries'][i]['summary'])
        docList.append(wordList)
        fullText.extend(wordList)
        classList.append(0)
    vocabList = createVocabList(docList)
    top30Words = calcMostFreq(vocabList, fullText)
    for pairW in top30Words:
        if pairW[0] in vocabList:
            vocabList.remove(pairW[0])
    trainingSet = range(2*minLen)
    testSet=[]
    for i in range(20):
        randIndex = int(random.uniform(0, len(trainingSet)))
        testSet.append(trainingSet[randIndex])
        del(trainingSet[randIndex])
    trainMat=[]; trainClasses=[]
    for docIndex in trainingSet:
        trainMat.append(bagOfWords2VecMN(vocabList, docList[docIndex]))
        trainClasses.append(classList[docIndex])
    p0V, p1V, pSpam = trainNB0(array(trainMat), array(trainClasses))
    errorCount = 0
    for docIndex in testSet:
        wordVector = bagOfWords2VecMN(vocabList, docList[docIndex])
        if classifyNB(array(wordVector), p0V, p1V, pSpam) != classList[docIndex]:
            errorCount += 1
    print 'the error rate is: ', float(errorCount)/len(testSet)
    return vocabList, p0V, p1V





if __name__=="__main__":
    listOPosts, listClasses = loadDataSet()
    print listOPosts, listClasses
    myVocabList = createVocabList(listOPosts)
    print myVocabList
    print setOfWord2Vec(myVocabList, listOPosts[0])
    trainMat = []
    for postinDoc in listOPosts:
        trainMat.append(setOfWord2Vec(myVocabList, postinDoc))
    p0V, p1V, pAb = trainNB0(trainMat, listClasses)
    print p0V
    print testingNB()
    spamTest()
    import feedparser
    ny = feedparser.parse('http://newyork.craigslist.org/stp/index.rss')
    sf = feedparser.parse('http://sfbay.craigslist.org/stp/index.rss')
    vocabList,pSF,pNY=localWords(ny,sf)