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) P(X,Y) ,然后求得后验概率分布 𝑃(𝑌|𝑋) P(Y|X) 。具体来说,利用训练数据学习 𝑃(𝑋|𝑌) P(X|Y) 和 𝑃(𝑌) P(Y) 的估计,得到联合概率分布:
𝑃(𝑋,𝑌)=𝑃(𝑌)𝑃(𝑋|𝑌)
概率估计方法可以是极大似然估计或贝叶斯估计。
2.朴素贝叶斯法的基本假设是条件独立性
这是一个较强的假设。由于这一假设,模型包含的条件概率的数量大为减少,朴素贝叶斯法的学习与预测大为简化。因而朴素贝叶斯法高效,且易于实现。其缺点是分类的性能不一定很高。
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。
将输入 𝑥 分到后验概率最大的类 𝑦 。
后验概率最大等价于0-1损失函数时的期望风险最小化。
模型: 高斯模型 多项式模型 伯努利模型
import numpy as npimport pandas as pdimport matplotlib.pyplot as plt%matplotlib inlinefrom sklearn.datasets import load_irisfrom sklearn.model_selection import train_test_splitfrom collections import Counterimport math# datadef create_data():iris = load_iris()df = pd.DataFrame(iris.data, columns=iris.feature_names)df['label'] = iris.targetdf.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]
class NaiveBayes:def __init__(self):self.model = None# 数学期望@staticmethoddef 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_traindef 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] = 1for 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 labeldef score(self, X_test, y_test):right = 0for X, y in zip(X_test, y_test):label = self.predict(X)if label == y:right += 1return right / float(len(X_test))
scikit-learn实例
from sklearn.naive_bayes import GaussianNBclf = GaussianNB() clf.fit(X_train, y_train)clf.score(X_test, y_test)clf.predict([[4.4, 3.2, 1.3, 0.2]])
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