第四章 朴素贝叶斯

基本梳理

• 朴素贝叶斯法的学习与分类
  • 基本方法
    • 训练集T={(x1,y1),(x2,y2),⋯ ,(xN,yN)}T = \left\{ \left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right) , \cdots , \left( x _ { N } , y _ { N } \right) \right\}T={(x1​,y1​),(x2​,y2​),⋯,(xN​,yN​)}
    • 由X和Y的联合概率分布P(X,Y)独立同分布产生
    • 朴素贝爷斯通过训练集学习联合概率分布P(X,Y)
      • 先验概率分布$P\left(Y=c_k\right),k=1,2,\cdots ,K$
      • 条件概率分布$P(X=x|Y=c_k)=P\left(X^{(1)}=x^{(1)},\cdots ,X^{(n)}=x^{(n)}|Y=c_k\right),k=1,2,\cdots ,K$
      • 条件独立性假设$\begin{array}{rl}P(X=x|Y=c_k)& =P\left(X^{(1)}=x^{(1)},\cdots ,X^{(n)}=x^{(n)}|Y=c_k\right)\\ & =\prod _{j=1}^{n}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)\end{array}$
      • 贝叶斯定理$P\left(Y=c_k|X=x\right)=\frac{P(X=x|Y=c_k)P\left(Y=c_k\right)}{{\sum }_{k}P(X=x|Y=c_k)P\left(Y=c_k\right)}$
      • 代入上式$P\left(Y=c_k|X=x\right)=\frac{P\left(Y=c_k\right){\prod }_{j}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)}{{\sum }_{k}P\left(Y=c_k\right){\prod }_{j}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)}$
      • 贝叶斯分类器$y=f(x)=\arg {\max }_{c_k}\frac{P\left(Y=c_k\right){\prod }_{j}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)}{{\sum }_{k}P\left(Y=c_k\right){\prod }_{j}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)}$
      • 分母对所有$c_k$都相同$y=\arg {\max }_{\genfrac{}{}{0ex}{}{a}{q}}P\left(Y=c_k\right){\prod }_{j}P\left(X^{(U)}=x^{(j)}|Y=c_k\right)$
• 朴素贝叶斯法的参数估计
  • 应用极大似然估计法估计相应的概率
  • 先验概率$\mathrm{P}\left(\mathrm{Y}={\mathrm{c}}_{\mathrm{k}}\right)$的极大似然估计是$P\left(Y=c_k\right)=\frac{{\sum }_{i=1}^{N}I\left(y_i=c_k\right)}{N},k=1,2,\cdots ,K$
  • 设第j个特征${\mathbf{X}}^{(j)}$可能取值的集合为:$\left\{a_{j1},a_{j2},\cdots ,a_{j{s}_j}\right\}$
  • 条件概率的极大似然估计:$P\left(X^{(j)}=a_{jl}|Y=c_k\right)=\frac{{\sum }_{i=1}^{N}I\left(x_i^{(j)}=a_{jl}{y}_i=c_k\right)}{{\sum }_{i=1}^{N}I\left(y_i=c_k\right)}$
  • 步骤
    • 计算先验概率和条件概率
      • $P\left(Y=c_k\right)=\frac{{\sum }_{i=1}^{N}I\left(y_i=c_k\right)}{N},k=1,2,\cdots ,K$
      • $P\left(X^{(j)}=a_n|Y=c_k\right)=\frac{{\sum }_{i=1}^{N}I\left(x_i^{(j)}=a_j,y_i=c_k\right)}{{\sum }_{i=1}^{N}I\left(y_i=c_k\right)}$
      • $j=1,2,\cdots ,n;l=1,2,\cdots ,S_j;k=1,2,\cdots ,K$
    • 对于给定的实例$x={\left(x^{(1)},x^{(2)},\cdots ,x^{(n)}\right)}^T$
      • 计算$P\left(Y=c_k\right){\prod }_{j=1}^{n}P\left(X^{(\prime )}=x^{(j)}|Y=c_k\right),k=1,2,\cdots ,K$
    • 确定x的类别
      • $y=\arg {\max }_{c_k}P\left(Y=c_k\right){\prod }_{j=1}^{n}P\left(X^{(j)}=x^{(j)}|Y=c_k\right)$

代码小练习

基于贝叶斯定理与特征条件独立假设的分类方法。

模型：

• 高斯模型
• 多项式模型
• 伯努利模型

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline

from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split

from collections import Counter
import math

创建数据

# data
def create_data():
    iris = load_iris()
    df = pd.DataFrame(iris.data, columns=iris.feature_names)
    df['label'] = iris.target
    df.columns = ['sepal length', 'sepal width', 'petal length', 'petal width', 'label']
    data = np.array(df.iloc[:100, :])
    # print(data)
    return data[:,:-1], data[:,-1]
X, y = create_data()
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3) 

创建模型

GaussianNB 高斯朴素贝叶斯

特征的可能性被假设为高斯

概率密度函数：
$$P(x_i|y_k)=\frac{1}{\sqrt{2\pi {\sigma }_{yk}^2}}exp(-\frac{(x_i-{\mu }_{yk}{)}^2}{2{\sigma }_{yk}^2})$$

数学期望(mean)：$\mu$，方差：${\sigma }^2=\frac{\sum (X-\mu {)}^2}{N}$

class NaiveBayes:
    def __init__(self):
        self.model = None
    # 数学期望
    @staticmethod
    def mean(X):
        return sum(X) / float(len(X))
    # 标准差
    def stdev(self,X):
        avg = self.mean(X)
        return math.sqrt(sum([pow(x-avg,2) for x in X])/float(len(X)))
    
    # 处理X_train
    def summarize(self,train_data):
        summarizes = [(self.mean(i),self.stdev(i)) for i in zip(*train_data)]
        return summarizes
    
    # 概率密度函数
    def gaussian_probability(self,x,mean,stdev):
        exponent = math.exp(-(math.pow(x-mean,2)/(2*math.pow(stdev,2))))
        return (1/(math.sqrt(2*math.pi)*stdev))*exponent
    # 分类别求出数学期望和标准差
    def fit(self, X, y):
        labels = list(set(y))
        data = {label:[] for label in labels}
        for f, label in zip(X, y):
            data[label].append(f)
#         print("分类",data)
        self.model = {label : self.summarize(value) for label,value in data.items()}
#         print("分类后期望和方差",self.model)
        return "gaussianNB train done!"
    
    # 计算概率
    def calculate_probabilities(self,input_data):
        probabilities = {}
        for label, value in self.model.items():
            probabilities[label] = 1
            for i in range(len(value)):
                mean, stdev = value[i]
                probabilities[label] *= self.gaussian_probability(input_data[i],mean,stdev)
        return probabilities
    # 类别
    def predict(self,X_test):
        label = sorted(self.calculate_probabilities(X_test).items(),key = lambda x : x[-1])[-1][0]
        return label
    def score(self,X_test,y_test):
        right = 0
        for X,y in zip(X_test,y_test):
            label = self.predict(X)
            if label == y:
                right += 1
        return right/float(len(X_test))

model = NaiveBayes()

model.fit(X_train, y_train)

'gaussianNB train done!'

print(model.predict([4.4,  3.2,  1.3,  0.2]))

0.0

model.score(X_test, y_test)

1.0

scikit-learn实例

scikit-learn实例

from sklearn.naive_bayes import GaussianNB

clf = GaussianNB()
clf.fit(X_train, y_train)

GaussianNB(priors=None)

clf.score(X_test, y_test)

1.0

clf.predict([[4.4,  3.2,  1.3,  0.2]])

array([ 0.])

from sklearn.naive_bayes import BernoulliNB, MultinomialNB# 伯努利模型和多项式模型

clf2 = BernoulliNB()
clf2.fit(X_train,y_train)

BernoulliNB(alpha=1.0, binarize=0.0, class_prior=None, fit_prior=True)

clf2.score(X_test, y_test)

0.46666666666666667

clf.predict([[4.4,  3.2,  1.3,  0.2]])

array([ 0.])

clf3 = MultinomialNB()
clf3.fit(X_train,y_train)

MultinomialNB(alpha=1.0, class_prior=None, fit_prior=True)

clf3.score(X_test, y_test)

1.0

clf3.predict([[4.4,  3.2,  1.3,  0.2]])

array([ 0.])