朴素贝叶斯分类器（MNIST数据集）

P(y|X)=P(y)*P(X|y)/P(X)

样本中的属性相互独立；

 

原问题的等价问题为：

 

数据处理
为防止P(y)*P(X|y)的值下溢，对原问题取对数，即：

 

 

 

注意：若某属性值在训练集中没有与某个类同时出现过，则直接P(y)或P(X|y)可能为0，这样计算出P(y)*P(X|y)的值为0，没有可比性，且不便于求对数，因此需要对概率值进行“平滑”处理，常用拉普拉斯修正。

先验概率修正：令Dy表示训练集D中第y类样本组合的集合，N表示训练集D中可能的类别数

 

即每个类别的样本个数都加 1。

类条件概率：另Dy,xi表示Dc中在第 i 个属性上取值为xi的样本组成的集合，Ni表示第 i 个属性可能的取值数

 

即该类别中第 i 个属性都增加一个样本。

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数据预处理

 

训练模型

测试样本

函数调用

 

 

 

 

参考

python朴素贝叶斯分类MNIST数据集

import struct
from numpy import *
import numpy as np
import time
def read_image(file_name):
    #先用二进制方式把文件都读进来
    file_handle=open(file_name,"rb")  #以二进制打开文档
    file_content=file_handle.read()   #读取到缓冲区中
    offset=0
    head = struct.unpack_from('>IIII', file_content, offset)  # 取前4个整数，返回一个元组
    offset += struct.calcsize('>IIII')
    imgNum = head[1]  #图片数
    rows = head[2]   #宽度
    cols = head[3]  #高度
 
    images=np.empty((imgNum , 784))#empty，是它所常见的数组内的所有元素均为空，没有实际意义，它是创建数组最快的方法
    image_size=rows*cols#单个图片的大小
    fmt='>' + str(image_size) + 'B'#单个图片的format
 
    for i in range(imgNum):
        images[i] = np.array(struct.unpack_from(fmt, file_content, offset))
        # images[i] = np.array(struct.unpack_from(fmt, file_content, offset)).reshape((rows, cols))
        offset += struct.calcsize(fmt)
    return images
 
#读取标签
def read_label(file_name):
    file_handle = open(file_name, "rb")  # 以二进制打开文档
    file_content = file_handle.read()  # 读取到缓冲区中
 
    head = struct.unpack_from('>II', file_content, 0)  # 取前2个整数，返回一个元组
    offset = struct.calcsize('>II')
 
    labelNum = head[1]  # label数
    # print(labelNum)
    bitsString = '>' + str(labelNum) + 'B'  # fmt格式：'>47040000B'
    label = struct.unpack_from(bitsString, file_content, offset)  # 取data数据，返回一个元组
    return np.array(label)
 
def loadDataSet():
    #mnist
    train_x_filename="train-images-idx3-ubyte"
    train_y_filename="train-labels-idx1-ubyte"
    test_x_filename="t10k-images-idx3-ubyte"
    test_y_filename="t10k-labels-idx1-ubyte"
 
    # #fashion mnist
    # train_x_filename="fashion-train-images-idx3-ubyte"
    # train_y_filename="fashion-train-labels-idx1-ubyte"
    # test_x_filename="fashion-t10k-images-idx3-ubyte"
    # test_y_filename="fashion-t10k-labels-idx1-ubyte"
 
    train_x=read_image(train_x_filename)#60000*784 的矩阵
    train_y=read_label(train_y_filename)#60000*1的矩阵
    test_x=read_image(test_x_filename)#10000*784
    test_y=read_label(test_y_filename)#10000*1
 
    train_x=normalize(train_x)
    test_x=normalize(test_x)
    # #调试的时候让速度快点，就先减少数据集大小
    # train_x=train_x[0:1000,:]
    # train_y=train_y[0:1000]
    # test_x=test_x[0:500,:]
    # test_y=test_y[0:500]
 
    return train_x, test_x, train_y, test_y
 
def  normalize(data):#图片像素二值化，变成0-1分布
    m=data.shape[0]
    n=np.array(data).shape[1]
    for i in range(m):
        for j in range(n):
            if data[i,j]!=0:
                data[i,j]=1
            else:
                data[i,j]=0
    return data
 
#（1）计算先验概率及条件概率
def train_model(train_x,train_y,classNum):#classNum是指有10个类别，这里的train_x是已经二值化，
    m=train_x.shape[0]
    n=train_x.shape[1]
    # prior_probability=np.zeros(n)#先验概率
    prior_probability=np.zeros(classNum)#先验概率
    conditional_probability=np.zeros((classNum,n,2))#条件概率
    #计算先验概率和条件概率
    for i in range(m):#m是图片数量，共60000张
        img=train_x[i]#img是第i个图片，是1*n的行向量
        label=train_y[i]#label是第i个图片对应的label
        prior_probability[label]+=1#统计label类的label数量(p(Y=ck)，下标用来存放label,prior_probability[label]除以n就是某个类的先验概率
        for j in range(n):#n是特征数，共784个
            temp=img[j].astype(int)#img[j]是0.0，放到下标去会显示错误，只能用整数
 
            conditional_probability[label][j][temp] += 1
 
            # conditional_probability[label][j][img[j]]+=1#统计的是类为label的，在每个列中为1或者0的行数为多少，img[j]的值要么就是0要么就是1，计算条件概率
 
    #将概率归到[1.10001]
    for i in range(classNum):
        for j in range(n):
            #经过二值化的图像只有0，1两种取值
            pix_0=conditional_probability[i][j][0]
            pix_1=conditional_probability[i][j][1]
 
            #计算0，1像素点对应的条件概率
            probability_0=(float(pix_0)/float(pix_0+pix_1))*10000+1
            probability_1 = (float(pix_1)/float(pix_0 + pix_1)) * 10000 + 1
 
            conditional_probability[i][j][0]=probability_0
            conditional_probability[i][j][1]=probability_1
    return prior_probability,conditional_probability
 
#（2）对给定的x，计算先验概率和条件概率的乘积
def cal_probability(img,label,prior_probability,conditional_probability):
    probability=int(prior_probability[label])#先验概率
    n=img.shape[0]
    # print(n)
    for i in range(n):#应该是特征数
        probability*=int(conditional_probability[label][i][img[i].astype(int)])
 
    return probability
 
#确定实例x的类，相当于argmax
def predict(test_x,test_y,prior_probability,conditional_probability):#传进来的test_x或者是train_x都是二值化后的
    predict_y=[]
    m=test_x.shape[0]
    n=test_x.shape[1]
    for i in range(m):
        img=np.array(test_x[i])#img已经是二值化以后的列向量
        label=test_y[i]
        max_label=0
        max_probability= cal_probability(img,0,prior_probability,conditional_probability)
        for j in range(1,10):#从下标为1开始，因为初始值是下标为0
            probability=cal_probability(img,j,prior_probability,conditional_probability)
            if max_probability<probability:
                max_probability=probability
                max_label=j
        predict_y.append(max_label)#用来记录每行最大概率的label
    return np.array(predict_y)
 
def cal_accuracy(test_y,predict_y):
    m=test_y.shape[0]
    errorCount=0.0
    for i in range(m):
        if test_y[i]!=predict_y[i]:
            errorCount+=1
    accuracy=1.0-float(errorCount)/m
    return accuracy
 
if __name__=='__main__':
    classNum=10
    print("Start reading data...")
    time1=time.time()
    train_x, test_x, train_y, test_y=loadDataSet()
    train_x=normalize(train_x)
    test_x=normalize(test_x)
 
    time2=time.time()
    print("read data cost",time2-time1,"second")
 
    print("start training data...")
    prior_probability, conditional_probability=train_model(train_x,train_y,classNum)
    for i in range(classNum):
        print(prior_probability[i])#输出一下每个标签的总共数量
    time3=time.time()
    print("train data cost",time3-time2,"second")
 
    print("start predicting data...")
    predict_y=predict(test_x,test_y,prior_probability,conditional_probability)
    time4=time.time()
    print("predict data cost",time4-time3,"second")
 
    print("start calculate accuracy...")
    acc=cal_accuracy(test_y,predict_y)
    time5=time.time()
    print("accuarcy",acc)
    print("calculate accuarcy cost",time5-time4,"second")