朴素贝叶斯算法学习学习笔记_贝利叶算法-优快云博客

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朴素贝叶斯算法学习笔记

~~本文仅为了个人学习理解使用~~

朴素贝叶斯分类器（是一种分类方法）

贝叶斯公式

$\mid B)=\frac{P(A,B) }{P(B)}=\frac{P(B \mid A) P(A)}{P(B)}$
其中：
$P (A)$ :先验概率
$\mid B)$ :后验概率
$\mid A)$ :在事件A发生的条件下B发生的概率，即似然函数
$P (B)$ :对于所有类标记都相同，所以证据因子 $P (B)$ 与类标记无关
$P\left(A_{i} \mid B\right)=\frac{P\left(B \mid A_{i}\right) P\left(A_{i}\right)}{\sum_{j} P\left(B \mid A_{j}\right) P\left(A_{j}\right)}$

分类问题

朴素贝叶斯算法（带例题解释）
分类问题
则贝叶斯公式可以表示为：
$P\left(y=c_{n} \mid x=X\right)=\frac{P\left(x=X, y=c_{n}\right)}{P(x=X)}=\frac{P\left(x=X \mid y=c_{n}\right) P\left(y=c_{n}\right)}{P(x=X)}$
假设各个属性相互独立则：
$P\left(x=X \mid y=c_{n}\right)=\prod_{i=1}^{m} P\left(x^{i}=a_{i} \mid y=c_{n}\right)$
朴素贝叶斯公式：在特征集合 $x$ 的条件下， $y$ 取不同值的概率。
$P\left(y=c_{n} \mid x=X \right)=\frac{P\left(y=c_{n}\right) \prod_{i=1}^{m} P\left(x^{i}=a_{i} \mid y=c_{n}\right)}{P(x=X)}$
将使得条件概率最大 $y$ 作为预测结果：
$f(x)=\operatorname{argmax}\left(\frac{P\left(y=c_{n}\right) \prod_{c_{n}}^{m} P\left(x^{i}=a_{i} \mid y=c_{n}\right)}{P(x=X)}\right)$
（证据因子与类标记无关）即 $p (x = X)$ 可以省略
$f(x)=\arg \max _{c_{n}}\left(P\left(y=c_{n}\right) \prod_{i=1}^{m} P\left(x^{i}=a_{i} \mid y=c_{n}\right)\right)$
$P\left(y=c_{n}\right)=\frac{\sum_{i=1}^{N} I\left(y=c_{n}\right)}{N}, n=1,2, \ldots K$
$P\left(x^{i}=a_{j} \mid y=c_{n}\right)=\frac{\sum_{i=1}^{N} I\left(x_{i}^{j}=a_{j} \mid y=c_{n}\right)}{\sum_{i=1}^{N} I\left(y_{i}=c_{n}\right)}$

算例1

朴素贝叶斯算法的代码实例实现（python）

价格A	课时B	销量C	价格A	课时B	销量C
低	多	高	0	2	2
高	中	高	2	1	2
低	少	高	0	0	2
低	中	低	0	1	0
中	中	中	1	1	1
高	多	高	2	2	2
低	少	中	0	0	1

预测价格A=2（高）课时B=2（多）时销量

from __future__ import division
from numpy import array


def set_data(price, time, sale):
    price_number = []
    time_number = []
    sale_number = []
    for i in price:
        if i == "低":
            price_number.append(0)
        elif i == "中":
            price_number.append(1)
        elif i == "高":
            price_number.append(2)
    for j in time:
        if j == "少":
            time_number.append(0)
        elif j == "中":
            time_number.append(1)
        elif j == "多":
            time_number.append(2)
    for k in sale:
        if k == "低":
            sale_number.append(0)
        elif k == "中":
            sale_number.append(1)
        elif k == "高":
            sale_number.append(2)
    return price_number, time_number, sale_number

price = ["低", "高", "低", "低", "中", "高", "低"]
time = ["多", "中", "少", "中", "中", "多", "少"]
sale = ["高", "高", "高", "低", "中", "高", "中"]

price_number, time_number, sale_number = set_data(price, time, sale)
print(price_number, time_number, sale_number)

$P(C=0\mid x=X) \propto P(C=0)P(A=2 \mid C=0)P(B=2 \mid C=0)$
$P(C=1\mid x=X)\propto P(C=1)P(A=2 \mid C=1)P(B=2 \mid C=1)$
$P(C=2\mid x=X)\propto P(C=2)P(A=2 \mid C=2)P(B=2 \mid C=2)$

from __future__ import division

price_number = [0, 2, 0, 0, 1, 2, 0]
time_number = [2, 1, 0, 1, 1, 2, 0]
sale_number = [2, 2, 2, 0, 1, 2, 1]
exprice_number = 2
extime_number = 2

sale0p = sale_number.count(0)
sale1p = sale_number.count(1)
sale2p = sale_number.count(2)

a0 = 0
a1 = 0
a2 = 0
b0 = 0
b1 = 0
b2 = 0

for i in range(0, len(sale_number)):
    if price_number[i] == 2:
        if sale_number[i] == 0:
            a0 = a0 + 1
        elif sale_number[i] == 1:
            a1 = a1 + 1
        elif sale_number[i] == 2:
            a2 = a2 + 1

    if time_number[i] == 2:
        if sale_number[i] == 0:
            b0 = b0 + 1
        elif sale_number[i] == 1:
            b1 = b1 + 1
        elif sale_number[i] == 2:
            b2 = b2 + 1

pa0 = a0 / sale0p
pa1 = a1 / sale1p
pa2 = a2 / sale2p

pb0 = b0 / sale0p
pb1 = b1 / sale1p
pb2 = b2 / sale2p

pc0 = sale0p / len(sale_number)
pc1 = sale1p / len(sale_number)
pc2 = sale2p / len(sale_number)

pcc0 = pc0 * pa0 * pb0
pcc1 = pc1 * pa1 * pb1
pcc2 = pc2 * pa2 * pb2
indf = (pcc0, pcc1, pcc2)
print(indf)
max_indf = indf.index(max(indf))

if max_indf == 0:
    print('销量低')
elif max_indf == 1:
    print('销量中')
elif max_indf == 2:
    print('销量高')