# 四、实验记录

1．朴素贝叶斯法是典型的生成学习方法。生成方法由训练数据学习联合概率分布
$$P(X,Y)$$，然后求得后验概率分布$$P(Y|X)$$。具体来说，利用训练数据学习$$P(X|Y)$$$$P(Y)$$的估计，得到联合概率分布：

$P(X,Y)＝P(Y)P(X|Y)$

2．朴素贝叶斯法的基本假设是条件独立性，

\begin{aligned} 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{aligned}

3．朴素贝叶斯法利用贝叶斯定理与学到的联合概率模型进行分类预测。

$P(Y | X)=\frac{P(X, Y)}{P(X)}=\frac{P(Y) P(X | Y)}{\sum_{Y} P(Y) P(X | Y)}$

$y=\arg \max _{c_{k}} P\left(Y=c_{k}\right) \prod_{j=1}^{n} P\left(X_{j}=x^{(j)} | Y=c_{k}\right)$

• 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.model_selection import train_test_split

from collections import Counter
import math

# data
def create_data():
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)

X_test[0], y_test[0]



$P(x_i | y_k)=\frac{1}{\sqrt{2\pi\sigma^2_{yk}}}exp(-\frac{(x_i-\mu_{yk})^2}{2\sigma^2_{yk}})$

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（方差）
def stdev(self, X):
avg = self.mean(X)
return math.sqrt(sum([pow(x - avg, 2) for x in X]) / float(len(X)))

# 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

# 处理X_train
def summarize(self, train_data):
summaries = [(self.mean(i), self.stdev(i)) for i in zip(*train_data)]
return summaries

# 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)
self.model = {
label: self.summarize(value)
for label, value in data.items()
}
return ‘gaussianNB train done!‘

# 计算概率
def calculate_probabilities(self, input_data):
# summaries:{0.0: [(5.0, 0.37),(3.42, 0.40)], 1.0: [(5.8, 0.449),(2.7, 0.27)]}
# input_data:[1.1, 2.2]
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):
# {0.0: 2.9680340789325763e-27, 1.0: 3.5749783019849535e-26}
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)
print(model.predict([4.4,  3.2,  1.3,  0.2]))
model.score(X_test, y_test)


from sklearn.naive_bayes import GaussianNB
clf = GaussianNB()
clf.fit(X_train, y_train)
clf.score(X_test, y_test)
clf.predict([[4.4,  3.2,  1.3,  0.2]])
from sklearn.naive_bayes import BernoulliNB, MultinomialNB # 伯努利模型和多项式模型


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