朴素贝叶斯算法简介及python代码实现分析

概念：

　　贝叶斯定理：贝叶斯理论是以18世纪的一位神学家托马斯.贝叶斯(Thomas Bayes)命名。通常，事件A在事件B（发生）的条件下的概率，与事件B在事件A（发生）的条件下的概率是不一样的；然而，这两者是有确定的关系的，贝叶斯定理就是这种关系的陈述

　　朴素贝叶斯：朴素贝叶斯方法是基于贝叶斯定理和特征条件独立假设的分类方法。对于给定的训练数据集，首先基于特征条件独立假设学习输入/输出的联合概率分布；然后基于此模型，对给定的输入x，利用贝叶斯定理求出后验概率(Maximum A Posteriori)最大的输出y。

通俗的来讲，在给定数据集的前提下，对于一个新样本（未分类），在数据集中找到和新样本特征相同的样本，最后根据这些样本算出每个类的概率，概率最高的类即为新样本的类。

运算公式：

P( h | d) = P ( d | h ) * P( h) / P(d)

这里：
P ( h | d )：是因子h基于数据d的假设概率,叫做后验概率
P ( d | h ) : 是假设h为真条件下的数据d的概率
P( h)　: 是假设条件h为真的时候的概率（和数据无关），它叫做h的先验概率
P(d)　: 数据d的概率，和先验条件无关．

算法实现分解：

１　数据处理：加载数据并把他们分成训练数据和测试数据
２　汇总数据：汇总训练数据的概率以便后续计算概率和做预测
３　结果预测：　通过给定的测试数据和汇总的训练数据做预测
４　评估准确性：使用测试数据来评估预测的准确性

代码实现：

 # Example of Naive Bayes implemented from Scratch in Python
 import csv
 import random
 import math

 def loadCsv(filename):
         lines = csv.reader(open(filename, "rb"))
         dataset = list(lines)
         for i in range(len(dataset)):
                 dataset[i] = [float(x) for x in dataset[i]]
         return dataset

 def splitDataset(dataset, splitRatio):
         trainSize = int(len(dataset) * splitRatio)
         trainSet = []
         copy = list(dataset)
         while len(trainSet) < trainSize:
                 index = random.randrange(len(copy))
                 trainSet.append(copy.pop(index))
         return [trainSet, copy]

 def separateByClass(dataset):
         separated = {}
         for i in range(len(dataset)):
                 vector = dataset[i]
                 if (vector[-1] not in separated):
                         separated[vector[-1]] = []
                 separated[vector[-1]].append(vector)
         return separated

 def mean(numbers):
         return sum(numbers)/float(len(numbers))

 def stdev(numbers):
         avg = mean(numbers)
         variance = sum([pow(x-avg,2) for x in numbers])/float(len(numbers)-1)
         return math.sqrt(variance)

 def summarize(dataset):
         summaries = [(mean(attribute), stdev(attribute)) for attribute in zip(*dataset)]
         del summaries[-1]
         return summaries

 def summarizeByClass(dataset):
         separated = separateByClass(dataset)
         summaries = {}
         for classValue, instances in separated.iteritems():
                 summaries[classValue] = summarize(instances)
         return summaries

 def calculateProbability(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

 def calculateClassProbabilities(summaries, inputVector):
         probabilities = {}
         for classValue, classSummaries in summaries.iteritems():
                 probabilities[classValue] = 1
                 for i in range(len(classSummaries)):
                         mean, stdev = classSummaries[i]
                         x = inputVector[i]
                         probabilities[classValue] *= calculateProbability(x, mean, stdev)
         return probabilities

 def predict(summaries, inputVector):
         probabilities = calculateClassProbabilities(summaries, inputVector)
         bestLabel, bestProb = None, -1
         for classValue, probability in probabilities.iteritems():
                 if bestLabel is None or probability > bestProb:
                         bestProb = probability
                         bestLabel = classValue
         return bestLabel

 def getPredictions(summaries, testSet):
         predictions = []
         for i in range(len(testSet)):
                 result = predict(summaries, testSet[i])
                 predictions.append(result)
         return predictions

 def getAccuracy(testSet, predictions):
         correct = 0
         for i in range(len(testSet)):
                 if testSet[i][-1] == predictions[i]:
                         correct += 1
         return (correct/float(len(testSet))) * 100.0

 def main():
         filename = 'pima-indians-diabetes.data.csv'
         splitRatio = 0.67
         dataset = loadCsv(filename)
         trainingSet, testSet = splitDataset(dataset, splitRatio)
         print('Split {0} rows into train={1} and test={2} rows').format(len(dataset), len(trainingSet), len(testSet))
         # prepare model
         summaries = summarizeByClass(trainingSet)
         # test model
         predictions = getPredictions(summaries, testSet)
         accuracy = getAccuracy(testSet, predictions)
         print('Accuracy: {0}%').format(accuracy)

 main()

pima-indians-diabetes.data.csv的下载地址：

https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv

参考文档：

１　https://en.wikipedia.org/wiki/Naive_Bayes_classifier

２　https://machinelearningmastery.com/naive-bayes-classifier-scratch-python/

３　https://machinelearningmastery.com/naive-bayes-for-machine-learning/