贝叶斯

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sklearn实战-乳腺癌细胞数据挖掘

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 Python例子

# -*- coding: utf-8 -*-
"""
Created on Mon Jul 24 10:40:14 2017

@author: toby
"""

# Import standard packages
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
import pandas as pd
import seaborn as sns
import os
 
# additional packages
import pymc as pm
from scipy.stats.mstats import mquantiles
 
# additional packages
import sys
sys.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 $alpha, eta$")
    plt.hist(beta_samples, histtype='stepfilled', bins=35, alpha=0.85,
             label=r"posterior of $eta$", 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
 
    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 
probability 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)

 https://www.youtube.com/watch?v=vlAdYt_8d9A

一、什么是贝叶斯推断

贝叶斯推断（Bayesian inference）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（Bayes' theorem）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1
def 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)

python风控评分卡建模和风控常识

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