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Python例子
# -*- coding: utf-8 -*-"""Created on Mon Jul 24 10:40:14 2017@author: toby"""# Import standard packagesimport numpy as npimport matplotlib.pyplot as pltfrom scipy import statsimport pandas as pdimport seaborn as snsimport os # additional packagesimport pymc as pmfrom scipy.stats.mstats import mquantiles # additional packagesimport syssys.path.append(os.path.join('..', '..', 'Utilities'))try:# Import formatting commands if directory "Utilities" is available from ISP_mystyle import setFonts, showData except ImportError:# Ensure correct performance otherwise def setFonts(*options): return def showData(*options): plt.show() return sns.set_context('poster') def logistic(x, beta, alpha=0): '''Logistic Function''' return 1.0 / (1.0 + np.exp(np.dot(beta, x) + alpha)) def getData(): '''Get and show the O-ring data''' inFile = 'challenger_data.csv' challenger_data = np.genfromtxt(inFile, skip_header=1, usecols=[0, 1], missing_values='NA', delimiter=',') # drop the NA values challenger_data = challenger_data[~np.isnan(challenger_data[:, 1])] temperature = challenger_data[:, 0] failureData = challenger_data[:, 1] # defect or not? return (temperature, failureData) def showAndSave(temperature, failures): '''Shows the input data, and saves the resulting figure''' # Plot it, as a function of tempature plt.figure() setFonts() sns.set_style('darkgrid') np.set_printoptions(precision=3, suppress=True) plt.scatter(temperature, failures, s=200, color="k", alpha=0.5) plt.yticks([0, 1]) plt.ylabel("Damage Incident?") plt.xlabel("Outside Temperature [F]") plt.title("Defects of the Space Shuttle O-Rings vs temperature") plt.tight_layout outFile = 'Challenger_ORings.png' showData(outFile) def mcmcSimulations(temperature, failures): '''Perform the MCMC-simulations''' # Define the prior distributions for alpha and beta # 'value' sets the start parameter for the simulation # The second parameter for the normal distributions is the "precision", # i.e. the inverse of the standard deviation np.random.seed(1234) beta = pm.Normal("beta", 0, 0.001, value=0) alpha = pm.Normal("alpha", 0, 0.001, value=0) # Define the model-function for the temperature @pm.deterministic def p(t=temperature, alpha=alpha, beta=beta): return 1.0 / (1. + np.exp(beta * t + alpha)) # connect the probabilities in `p` with our observations through a # Bernoulli random variable. observed = pm.Bernoulli("bernoulli_obs", p, value=failures, observed=True) # Combine the values to a model model = pm.Model([observed, beta, alpha]) # Perform the simulations map_ = pm.MAP(model) map_.fit() mcmc = pm.MCMC(model) mcmc.sample(120000, 100000, 2) # --- Show the resulting posterior distributions --- alpha_samples = mcmc.trace('alpha')[:, None] # best to make them 1d beta_samples = mcmc.trace('beta')[:, None] return(alpha_samples, beta_samples) def showSimResults(alpha_samples, beta_samples): '''Show the results of the simulations, and save them to an outFile''' plt.figure(figsize=(12.5, 6)) sns.set_style('darkgrid') setFonts(18) # Histogram of the samples: plt.subplot(211) plt.title(r"Posterior distributions of the variables $\alpha, \beta$") plt.hist(beta_samples, histtype='stepfilled', bins=35, alpha=0.85, label=r"posterior of $\beta$", color="#7A68A6", normed=True) plt.legend() plt.subplot(212) plt.hist(alpha_samples, histtype='stepfilled', bins=35, alpha=0.85, label=r"posterior of $\alpha$", color="#A60628", normed=True) plt.legend() outFile = 'Challenger_Parameters.png' showData(outFile) def calculateProbability(alpha_samples, beta_samples, temperature, failures): '''Calculate the mean probability, and the CIs''' # Calculate the probability as a function of time t = np.linspace(temperature.min() - 5, temperature.max() + 5, 50)[:, None] p_t = logistic(t.T, beta_samples, alpha_samples) mean_prob_t = p_t.mean(axis=0) # --- Calculate CIs --- # vectorized bottom and top 2.5% quantiles for "confidence interval" quantiles = mquantiles(p_t, [0.025, 0.975], axis=0) return (t, mean_prob_t, p_t, quantiles) def showProbabilities(linearTemperature, temperature, failures, mean_prob_t, p_t, quantiles): '''Show the posterior probabilities, and save the resulting figures''' # --- Show the probability curve ---- plt.figure(figsize=(12.5, 4)) setFonts(18) plt.plot(linearTemperature, mean_prob_t, lw=3, label="Average posterior\n \ probability of defect") plt.plot(linearTemperature, p_t[0, :], ls="--", label="Realization from posterior") plt.plot(linearTemperature, p_t[-2, :], ls="--", label="Realization from posterior") plt.scatter(temperature, failures, color="k", s=50, alpha=0.5) plt.title("Posterior expected value of probability of defect, plus realizations") plt.legend(loc="lower left") plt.ylim(-0.1, 1.1) plt.xlim(linearTemperature.min(), linearTemperature.max()) plt.ylabel("Probability") plt.xlabel("Temperature [F]") outFile = 'Challenger_Probability.png' showData(outFile) # --- Draw CIs --- setFonts() sns.set_style('darkgrid') plt.fill_between(linearTemperature[:, 0], *quantiles, alpha=0.7, color="#7A68A6") plt.plot(linearTemperature[:, 0], quantiles[0], label="95% CI", color="#7A68A6", alpha=0.7) plt.plot(linearTemperature, mean_prob_t, lw=1, ls="--", color="k", label="average posterior \nprobability of defect") plt.xlim(linearTemperature.min(), linearTemperature.max()) plt.ylim(-0.02, 1.02) plt.legend(loc="lower left") plt.scatter(temperature, failures, color="k", s=50, alpha=0.5) plt.xlabel("Temperature [F]") plt.ylabel("Posterior Probability Estimate") outFile = 'Challenger_CIs.png' showData(outFile) if __name__=='__main__': (temperature, failures) = getData() showAndSave(temperature, failures) (alpha, beta) = mcmcSimulations(temperature, failures) showSimResults(alpha, beta) (linearTemperature, mean_p, p, quantiles) = calculateProbability(alpha, beta, temperature, failures) showProbabilities(linearTemperature, temperature, failures, mean_p, p, quantiles)
一、什么是贝叶斯推断
贝叶斯推断（）是一种统计学方法，用来估计统计量的某种性质。
它是贝叶斯定理（）的应用。英国数学家托马斯·贝叶斯（Thomas Bayes）在1763年发表的一篇论文中，首先提出了这个定理。
叶斯推断与其他统计学推断方法截然不同。它建立在主观判断的基础上，也就是说，你可以不需要客观证据，先估计一个值，然后根据实际结果不断修正。正是因为它的主观性太强，曾经遭到许多统计学家的诟病。贝叶斯推断需要大量的计算，因此历史上很长一段时间，无法得到广泛应用。只有计算机诞生以后，它才获得真正的重视。人们发现，许多统计量是无法事先进行客观判断的，而互联网时代出现的大型数据集，再加上高速运算能力，为验证这些统计量提供了方便，也为应用贝叶斯推断创造了条件，它的威力正在日益显现。
六、【例子】假阳性问题
第二个例子是一个医学的常见问题，与现实生活关系紧密。
我们把P(A)称为"先验概率"（Prior probability），即在B事件发生之前，我们对A事件概率的一个判断。P(A|B)称为"后验概率"（Posterior probability），即在B事件发生之后，我们对A事件概率的重新评估。P(B|A)/P(B)称为"可能性函数"（Likelyhood），这是一个调整因子，使得预估概率更接近真实概率。
所以，条件概率可以理解成下面的式子：
后验概率 ＝ 先验概率 ｘ 调整因子
这就是贝叶斯推断的含义。我们先预估一个"先验概率"，然后加入实验结果，看这个实验到底是增强还是削弱了"先验概率"，由此得到更接近事实的"后验概率"。
在这里，如果"可能性函数"P(B|A)/P(B)>1，意味着"先验概率"被增强，事件A的发生的可能性变大；如果"可能性函数"=1，意味着B事件无助于判断事件A的可能性；如果"可能性函数"<1，意味着"先验概率"被削弱，事件A的可能性变小
已知某种疾病的发病率是0.001，即1000人中会有1个人得病。现有一种试剂可以检验患者是否得病，它的准确率是0.99，即在患者确实得病的情况下，它有99%的可能呈现阳性。它的误报率是5%，即在患者没有得病的情况下，它有5%的可能呈现阳性。现有一个病人的检验结果为阳性，请问他确实得病的可能性有多大？
假定A事件表示得病，那么P(A)为0.001。这就是"先验概率"，即没有做试验之前，我们预计的发病率。再假定B事件表示阳性，那么要计算的就是P(A|B)。这就是"后验概率"，即做了试验以后，对发病率的估计。
我们得到了一个惊人的结果，P(A|B)约等于0.019。也就是说，即使检验呈现阳性，病人得病的概率，也只是从0.1%增加到了2%左右。这就是所谓的"假阳性"，即阳性结果完全不足以说明病人得病。
为什么会这样？为什么这种检验的准确率高达99%，但是可信度却不到2%？答案是与它的误报率太高有关。（【习题】如果误报率从5%降为1%，请问病人得病的概率会变成多少？）
有兴趣的朋友，还可以算一下"假阴性"问题，即检验结果为阴性，但是病人确实得病的概率有多大。然后问自己，"假阳性"和"假阴性"，哪一个才是医学检验的主要风险？
贝叶斯分类器_代码实现
# -*- coding: utf-8 -*-"""Created on Thu Jan 12 10:02:45 2017@author: Administrator"""from numpy import *#导入数据，列表和评论，0表示非侮辱，1表示侮辱def loadDataSet(): postingList=[['my', 'dog', 'has', 'flea', 'problems', 'help', 'please'], ['maybe', 'not', 'take', 'him', 'to', 'dog', 'park', 'stupid'], ['my', 'dalmation', 'is', 'so', 'cute', 'I', 'love', 'him'], ['stop', 'posting', 'stupid', 'worthless', 'garbage'], ['mr', 'licks', 'ate', 'my', 'steak', 'how', 'to', 'stop', 'him'], ['quit', 'buying', 'worthless', 'dog', 'food', 'stupid'], ['this','is','a','good','movie'], ['this','movie','is','suck'], ['so','fantanstic'], ['fuck','the','actress']] classVec = [0,1,0,1,0,1,0,1,0,1] #1 is abusive, 0 not return postingList,classVec#得到一个列表，包含去重的单词 def createVocabList(dataSet): vocabSet = set([]) #create empty set for document in dataSet: vocabSet = vocabSet | set(document) #union of the two sets return list(vocabSet)#inputSet为测试语句，vocabList为32个去重后的单词列表#遍历输入语句的每个单词，例如第一个单词是my,如果my在vocabList里面，则returnVec对应值为1def setOfWords2Vec(vocabList, inputSet): returnVec = [0]*len(vocabList) for word in inputSet: #print("word:",word) if word in vocabList: returnVec[vocabList.index(word)] = 1 #else: print ("the word: %s is not in my Vocabulary!") % word return returnVec#输入参数为32个不重复的单词列表，六篇文章的二维列表，得到特征向量的矩阵（二维列表）def Matrix_featureVectors(): trainMat=[] for postinDoc in listOPosts: trainMat.append(setOfWords2Vec(myVocabList, postinDoc)) return trainMat def trainNB0(trainMatrix,trainCategory): numTrainDocs = len(trainMatrix) numWords = len(trainMatrix[0]) pAbusive = sum(trainCategory)/float(numTrainDocs) p0Num = ones(numWords); p1Num = ones(numWords) #change to ones() #如果其中一个概率值为0，则最后乘积为0，所以把分母改为2 p0Denom = 2.0; p1Denom = 2.0 #change to 2.0 for i in range(numTrainDocs): if trainCategory[i] == 1: p1Num += trainMatrix[i] p1Denom += sum(trainMatrix[i]) else: p0Num += trainMatrix[i] p0Denom += sum(trainMatrix[i]) #太多很小书相乘会遇到下溢问题，所以取对数 p1Vect = log(p1Num/p1Denom) #change to log() p0Vect = log(p0Num/p0Denom) #change to log() return p0Vect,p1Vect,pAbusive#分类器def classifyNB(vec2Classify, p0Vec, p1Vec, pClass1): p1 = sum(vec2Classify * p1Vec) + log(pClass1) #element-wise mult p0 = sum(vec2Classify * p0Vec) + log(1.0 - pClass1) if p1 > p0: return 1 else: return 0#测试数据 def testingNB(testEntry): thisDoc = array(setOfWords2Vec(myVocabList, testEntry)) print (testEntry,'classified as: ',classifyNB(thisDoc,p0V,p1V,pAb)) value=classifyNB(thisDoc,p0V,p1V,pAb) return value #导入数据，列表语句listOPosts,listClasses评价列表[0,1,0,1,0,1]listOPosts,listClasses = loadDataSet()#得到一个列表，包含去重的单词myVocabList = createVocabList(listOPosts)#输入参数为32个不重复的单词列表，六篇文章的二维列表，得到特征向量的矩阵（二维列表）trainMat=Matrix_featureVectors()#得到侮辱性文档的概率以及两个类别的概率向量p0V,p1V,pAb = trainNB0(array(trainMat),array(listClasses))#测试非侮辱语句testEntry0 = ['love', 'my', 'dalmation']#测试侮辱语句testEntry1 = ['stupid', 'garbage']#测试语句testEntry3=['fuck','the','movie']testingNB(testEntry0)testingNB(testEntry1)testingNB(testEntry3)
