概念：

　　貝葉斯定理：貝葉斯理論是以18世紀的一位神學家托馬斯.貝葉斯(Thomas Bayes)命名。通常，事件A在事件B（發生）的條件下的概率，與事件B在事件A（發生）的條件下的概率是不一樣的；然而，這兩者是有確定的關系的，貝葉斯定理就是這種關系的陳述

　　樸素貝葉斯：樸素貝葉斯方法是基於貝葉斯定理和特征條件獨立假設的分類方法。對於給定的訓練數據集，首先基於特征條件獨立假設學習輸入/輸出的聯合概率分布；然後基於此模型，對給定的輸入x，利用貝葉斯定理求出後驗概率(Maximum A Posteriori)最大的輸出y。

通俗的來講，在給定數據集的前提下，對於一個新樣本（未分類），在數據集中找到和新樣本特征相同的樣本，最後根據這些樣本算出每個類的概率，概率最高的類即為新樣本的類。

運算公式：

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

這裏：
P ( h | d )：是因子h基於數據d的假設概率,叫做後驗概率
P ( d | h ) : 是假設h為真條件下的數據d的概率
P( h)　: 是假設條件h為真的時候的概率（和數據無關），它叫做h的先驗概率
P(d)　: 數據d的概率，和先驗條件無關．

算法實現分解：

１　數據處理：加載數據並把他們分成訓練數據和測試數據
２　匯總數據：匯總訓練數據的概率以便後續計算概率和做預測
３　結果預測：　通過給定的測試數據和匯總的訓練數據做預測
４　評估準確性：使用測試數據來評估預測的準確性

代碼實現：

  1 # Example of Naive Bayes implemented from Scratch in Python
  2 import csv
  3 import random
  4 import math
  5 
  6 def loadCsv(filename):
  7         lines = csv.reader(open(filename, "rb"))
  8         dataset = list(lines)
  9         for i in range(len(dataset)):
 10                 dataset[i] = [float(x) for
 
 x in dataset[i]]
 11         return dataset
 12 
 13 def splitDataset(dataset, splitRatio):
 14         trainSize = int(len(dataset) * splitRatio)
 15         trainSet = []
 16         copy = list(dataset)
 17         while len(trainSet) < trainSize:
 18                 index = random.randrange(len(copy))
 19                 trainSet.append(copy.pop(index))
 20         return [trainSet, copy]
 21 
 22 def separateByClass(dataset):
 23         separated = {}
 24         for i in range(len(dataset)):
 25                 vector = dataset[i]
 26                 if (vector[-1] not in separated):
 27                         separated[vector[-1]] = []
 28                 separated[vector[-1]].append(vector)
 29         return separated
 30 
 31 def mean(numbers):
 32         return sum(numbers)/float(len(numbers))
 33 
 34 def stdev(numbers):
 35         avg = mean(numbers)
 36         variance = sum([pow(x-avg,2) for x in numbers])/float(len(numbers)-1)
 37         return math.sqrt(variance)
 38 
 39 def summarize(dataset):
 40         summaries = [(mean(attribute), stdev(attribute)) for attribute in zip(*dataset)]
 41         del summaries[-1]
 42         return summaries
 43 
 44 def summarizeByClass(dataset):
 45         separated = separateByClass(dataset)
 46         summaries = {}
 47         for classValue, instances in separated.iteritems():
 48                 summaries[classValue] = summarize(instances)
 49         return summaries
 50 
 51 def calculateProbability(x, mean, stdev):
 52         exponent = math.exp(-(math.pow(x-mean,2)/(2*math.pow(stdev,2))))
 53         return (1 / (math.sqrt(2*math.pi) * stdev)) * exponent
 54 
 55 def calculateClassProbabilities(summaries, inputVector):
 56         probabilities = {}
 57         for classValue, classSummaries in summaries.iteritems():
 58                 probabilities[classValue] = 1
 59                 for i in range(len(classSummaries)):
 60                         mean, stdev = classSummaries[i]
 61                         x = inputVector[i]
 62                         probabilities[classValue] *= calculateProbability(x, mean, stdev)
 63         return probabilities
 64 
 65 def predict(summaries, inputVector):
 66         probabilities = calculateClassProbabilities(summaries, inputVector)
 67         bestLabel, bestProb = None, -1
 68         for classValue, probability in probabilities.iteritems():
 69                 if bestLabel is None or probability > bestProb:
 70                         bestProb = probability
 71                         bestLabel = classValue
 72         return bestLabel
 73 
 74 def getPredictions(summaries, testSet):
 75         predictions = []
 76         for i in range(len(testSet)):
 77                 result = predict(summaries, testSet[i])
 78                 predictions.append(result)
 79         return predictions
 80 
 81 def getAccuracy(testSet, predictions):
 82         correct = 0
 83         for i in range(len(testSet)):
 84                 if testSet[i][-1] == predictions[i]:
 85                         correct += 1
 86         return (correct/float(len(testSet))) * 100.0
 87 
 88 def main():
 89         filename = ‘pima-indians-diabetes.data.csv‘
 90         splitRatio = 0.67
 91         dataset = loadCsv(filename)
 92         trainingSet, testSet = splitDataset(dataset, splitRatio)
 93         print(‘Split {0} rows into train={1} and test={2} rows‘).format(len(dataset), len(trainingSet), len(testSet))
 94         # prepare model
 95         summaries = summarizeByClass(trainingSet)
 96         # test model
 97         predictions = getPredictions(summaries, testSet)
 98         accuracy = getAccuracy(testSet, predictions)
 99         print(‘Accuracy: {0}%‘).format(accuracy)
100 
101 main()

pima-indians-diabetes.data.csv的下載地址：

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

參考文檔：

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/

樸素貝葉斯算法簡介及python代碼實現分析