貝葉斯定理

其中

所以樸素貝葉斯分類器為

因為分母是一定的，與Y無關，所以簡化為

樸素貝葉斯算法

貝葉斯估計

因為

可能為0，則無法計算

所以將樸素貝葉斯修正為貝葉斯估計

代碼實現

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

# 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)

#假設特征服從高斯分布
class NaiveBayes:
    def __init__(self):
        self.model = None

    # 數學期望
    @staticmethod
    def mean(X):
        return sum(X) / float(len(X))

    # 標準差（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)))

    # 概率密度函數
    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

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

    # 分類別求出數學期望和標準差
    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!'

    # 計算概率
    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

    # 類別
    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)

#sklearn實現
from sklearn.naive_bayes import GaussianNB
clf = GaussianNB()
clf.fit(X_train, y_train)

#假設特征服從伯努利模型和多項式模型
from sklearn.naive_bayes import BernoulliNB, MultinomialNB # 伯努利模型和多項式模型