贝叶斯定理
贝叶斯定理是通过对观测值概率分布的主观判断（即先验概率）进行修正的定理，在概率论中具有重要地位。
先验概率分布(边缘概率)是指基于主观判断而非样本分布的概率分布，后验概率(条件概率)是根据样本分布和未知参数的先验概率分布求得的条件概率分布。
贝叶斯公式：
P(A∩B) = P(A)*P(B|A) = P(B)*P(A|B)变形得：
P(A|B)=P(B|A)*P(A)/P(B)其中
P(A)是A的先验概率或边缘概率，称作"先验"是因为它不考虑B因素。P(A|B)是已知B发生后A的条件概率，也称作A的后验概率。P(B|A)是已知A发生后B的条件概率，也称作B的后验概率，这里称作似然度。P(B)是B的先验概率或边缘概率，这里称作标准化常量。P(B|A)/P(B)称作标准似然度。
朴素贝叶斯分类(Naive Bayes)
朴素贝叶斯分类器在估计类条件概率时假设属性之间条件独立。
首先定义
x = {a1,a2,...}为一个样本向量，a为一个特征属性div = {d1 = [l1,u1],...}特征属性的一个划分class = {y1,y2,...}样本所属的类别
算法流程：
(1) 通过样本集中类别的分布，对每个类别计算先验概率p(y[i])
(2) 计算每个类别下每个特征属性划分的频率p(a[j] in d[k] | y[i])
(3) 计算每个样本的p(x|y[i])
p(x|y[i]) = p(a[1] in d | y[i]) * p(a[2] in d | y[i]) * ...样本的所有特征属性已知，所以特征属性所属的区间d已知。
可以通过(2)确定p(a[k] in d | y[i])的值，从而求得p(x|y[i])。
(4) 由贝叶斯定理得：
p(y[i]|x) = ( p(x|y[i]) * p(y[i]) ) / p(x)因为分母相同，只需计算分子。
p(y[i]|x)是观测样本属于分类y[i]的概率，找出最大概率对应的分类作为分类结果。
示例：
导入数据集
{a1 = 0, a2 = 0, C = 0} {a1 = 0, a2 = 0, C = 1}
{a1 = 0, a2 = 0, C = 0} {a1 = 0, a2 = 0, C = 1}
{a1 = 0, a2 = 0, C = 0} {a1 = 0, a2 = 0, C = 1}
{a1 = 1, a2 = 0, C = 0} {a1 = 0, a2 = 0, C = 1}
{a1 = 1, a2 = 0, C = 0} {a1 = 0, a2 = 0, C = 1}
{a1 = 1, a2 = 0, C = 0} {a1 = 1, a2 = 0, C = 1}
{a1 = 1, a2 = 1, C = 0} {a1 = 1, a2 = 0, C = 1}
{a1 = 1, a2 = 1, C = 0} {a1 = 1, a2 = 1, C = 1}
{a1 = 1, a2 = 1, C = 0} {a1 = 1, a2 = 1, C = 1}
{a1 = 1, a2 = 1, C = 0} {a1 = 1, a2 = 1, C = 1}
计算类别的先验概率
P(C = 0) = 0.5
P(C = 1) = 0.5
计算每个特征属性条件概率：
P(a1 = 0 | C = 0) = 0.3
P(a1 = 1 | C = 0) = 0.7
P(a2 = 0 | C = 0) = 0.4
P(a2 = 1 | C = 0) = 0.6
P(a1 = 0 | C = 1) = 0.5
P(a1 = 1 | C = 1) = 0.5
P(a2 = 0 | C = 1) = 0.7
P(a2 = 1 | C = 1) = 0.3
测试样本：
x = { a1 = 1, a2 = 2}
p(x | C = 0) = p(a1 = 1 | C = 0) * p( 2 = 2 | C = 0) = 0.3 * 0.6 = 0.18
p(x | C = 1) = p(a1 = 1 | C = 1) * p (a2 = 2 | C = 1) = 0.5 * 0.3 = 0.15
计算P(C | x) * p(x):
P(C = 0) * p(x | C = 1) = 0.5 * 0.18 = 0.09
P(C = 1) * p(x | C = 2) = 0.5 * 0.15 = 0.075
所以认为测试样本属于类型C1
Python实现
朴素贝叶斯分类器的训练过程为计算(1),(2)中的概率表，应用过程为计算(3),(4)并寻找最大值。
还是使用原来的接口进行类封装：
from numpy import *
class NaiveBayesClassifier(object):
def __init__(self):
self.dataMat = list()
self.labelMat = list()
self.pLabel1 = 0
self.p0Vec = list()
self.p1Vec = list()
def loadDataSet(self,filename):
fr = open(filename)
for line in fr.readlines():
lineArr = line.strip().split()
dataLine = list()
for i in lineArr:
dataLine.append(float(i))
label = dataLine.pop() # pop the last column referring to label
self.dataMat.append(dataLine)
self.labelMat.append(int(label))
def train(self):
dataNum = len(self.dataMat)
featureNum = len(self.dataMat[0])
self.pLabel1 = sum(self.labelMat)/float(dataNum)
p0Num = zeros(featureNum)
p1Num = zeros(featureNum)
p0Denom = 1.0
p1Denom = 1.0
for i in range(dataNum):
if self.labelMat[i] == 1:
p1Num += self.dataMat[i]
p1Denom += sum(self.dataMat[i])
else:
p0Num += self.dataMat[i]
p0Denom += sum(self.dataMat[i])
self.p0Vec = p0Num/p0Denom
self.p1Vec = p1Num/p1Denom
def classify(self, data):
p1 = reduce(lambda x, y: x * y, data * self.p1Vec) * self.pLabel1
p0 = reduce(lambda x, y: x * y, data * self.p0Vec) * (1.0 - self.pLabel1)
if p1 > p0:
return 1
else:
return 0
def test(self):
self.loadDataSet(‘testNB.txt‘)
self.train()
print(self.classify([1, 2]))
if __name__ == ‘__main__‘:
NB = NaiveBayesClassifier()
NB.test()Matlab
Matlab的标准工具箱提供了对朴素贝叶斯分类器的支持：
trainData = [0 1; -1 0; 2 2; 3 3; -2 -1;-4.5 -4; 2 -1; -1 -3];
group = [1 1 -1 -1 1 1 -1 -1]‘;
model = fitcnb(trainData, group)
testData = [5 2;3 1;-4 -3];
predict(model, testData)fitcnb用来训练模型，predict用来预测。