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BAYES AND EMPIRICAL BAYES METHODS FOR DATA ANALYSIS

Bayesian

data

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由Bradley P. Carlin等人编著的Bayesian statistical learning方面的经典教材，覆盖了贝叶斯统计推断的系统理论，包括Bayesian inference method, empirical Bayes approaches, Bayesian computation algorithms, Model selection 以及 advanced Bayesian models 等章节.

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BAYES AND

EMPIRICAL

BAYES

METHODS

FOR

DATA ANALYSIS

Second edition

Bradley P. Carlin

Professor, Division of Biostatistics,

School of Public Health, University of Minnesota,

Minneapolis,

and

Thomas A. Louis

Senior Statistician,

The RAND Corporation,

Santa Monica, CA

CHAPMAN & HALL/CRC

A CRC Press Company

Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data

Carlin, Bradley P.

Bayes and empirical bayes methods for data analysis / Bradley P. Carlin and Thomas A.

Louis--2nd ed.



cm. (Texts in statistical science series)

Previous ed.: New York: Chapman & Hall, 1996.

Includes bibliographical references and indexes.

ISBN 58488-170-4

Bayesian statistical decision theory. I. Louis, Thomas A., II. Title. III. Texts

in statistical science

QA279.5.C36 2000

519'.5842-dc21



00-038242

CIP

This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic

or mechanical, including photocopying, microfilming, and recording, or by any information storage or

retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for

creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC

for such copying.

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at

www.crcpress.co

No claim to original U.S. Government works

International Standard Book Number 1-58488-170-4

Library of Congress Card Number 00-38242

Printed in the United States of America



3 4 5 6 7 8 9 0

Printed on acid-free paper

Contents

Preface to the Second Edition

Preface to the First Edition



Approaches for statistical inference

1.1

Introduction

1.2



Motivating vignettes

1.2.1



Personal probability

1.2.2

Missing data

1.2.3

Bioassay

1.2.4

Attenuation adjustment

1.3



Defining the approaches

1.4



The Bayes-frequentist controversy

1.5



Some basic Bayesian models

1.5.1



A Gaussian/Gaussian (normal/normal) model

1.5.2



A beta/binomial model

1.6

Exercises

The Bayes approach

2.1

Introduction

2.2



Prior distributions

2.2.1



Elicited priors

2.2.2



Conjugate priors

2.2.3



Noninformative priors

2.2.4

Other prior construction methods

2.3



Bayesian inference

2.3.1



Point estimation

2.3.2

Interval estimation

2.3.3



Hypothesis testing and Bayes factors

2.3.4

Example: Consumer preference data

2.4

Model assessment

2.4.1



Diagnostic measures

2.4.2

Model averaging

2.5

Nonparametric methods

2.6

Exercises

The empirical Bayes approach

3.1

Introduction

3.2



Nonparametric EB (NPEB) point estimation

3.2.1



Compound sampling models

3.2.2

Simple NPEB (Robbins' method)

3.2.3

Example: Accident data

3.3



Parametric EB (PEB) point estimation

3.3.1



Gaussian/Gaussian models

3.3.2



Beta/binomial model

3.3.3



EB performance of the PEB

3.3.4

Stein estimation

3.4



Computation via the EM algorithm

3.4.1



EM for PEB

3.4.2



Computing the observed information

3.4.3

EM for NPEB

3.4.4

Speeding convergence and generalizations

3.5



Interval estimation

3.5.1

Morris' approach

3.5.2



Marginal posterior approach

3.5.3

Bias correction approach

3.6



Generalization to regression structures

3.7

Exercises

Performance of Bayes procedures

4.1



Bayesian processing

4.1.1



Univariate stretching with a two-point prior

4.1.2



Multivariate Gaussian model

4.2



Frequentist performance: Point estimates

4.2.1



Gaussian/Gaussian model

4.2.2



Beta/binomial model

4.2.3

Generalization

4.3

Frequentist performance: Confidence intervals

4.3.1



Beta/binomial model

4.3.2

Fieller-Creasy problem

4.4



Empirical Bayes performance

4.4.1

Point estimation

4.4.2

Interval estimation

4.5



Design of experiments

4.5.1



Bayesian design for frequentist analysis

4.5.2



Bayesian design for Bayesian analysis

4.6

Exercises

Bayesian computation

5.1

Introduction

5.2



Asymptotic methods

5.2.1



Normal approximation

5.2.2



Laplace's method

5.3



Noniterative Monte Carlo methods

5.3.1

Direct sampling

5.3.2

Indirect

methods

5.4



Markov chain Monte Carlo methods

5.4.1



Substitution sampling and data augmentation

5.4.2



Gibbs sampling

5.4.3

Metropolis-Hastings algorithm

5.4.4



Hybrid forms and other algorithms

5.4.5

Variance estimation

5.4.6



Convergence monitoring and diagnosis

5.5

Exercises



Model criticism and selection

6.1



Bayesian robustness

6.1.1

Sensitivity analysis

6.1.2

Prior partitioning

6.2



Model assessment

6.3



Hayes factors via marginal density estimation

6.3.1

Direct methods

6.3.2



Using Gibbs sampler output

6.3.3



Using Metropolis-Hastings output

6.4



Bayes factors via sampling over the model space

6.4.1



Product space search

6.4.2



"Metropolized" product space search

6.4.3



Reversible jump MCMC

6.4.4



Using partial analytic structure

6.5



Other model selection methods

6.5.1



Penalized likelihood criteria

6.5.2



Predictive model selection

6.6

Exercises

Special methods and models

7.1



Estimating histograms and ranks

7.1.1



Model and inferential goals

7.1.2



Triple goal estimates

7.1.3



Smoothing and robustness

7.2



Order restricted inference

剩余431页未读，继续阅读

def main(): test_datasets = generate_attires.gen_attrs() classfier = bayes_classfier.navie_bayes_classifier() for data in test_datasets: print("特征值：", end='\t') print(data) print("预测结果：", end='\t') res = classfier.get_label(*data) # 表示多参传入 print(res) # 预测属于哪种水果的概率 print('水果类别：', end='\t') # 对后验概率排序，输出概率最大的标签 print(sorted(res.items(), key=operator.itemgetter(1), reverse=True)[0][0])这段代码写了什么

for data in test_datasets: # 输出特征值 print("特征值：", end='\t') print(data) # 预测结果 res = classfier.get_label(*data) # 表示多参传入 print("预测结果：", end='\t') print(res) # 预测属于...

分析完成def fit_naive_bayes_model(matrix, labels): """Fit a naive bayes model. This function should fit a Naive Bayes model given a training matrix and labels. The function should return the state of that model. Feel free to use whatever datatype you wish for the state of the model. Args: matrix: A numpy array containing word counts for the training data labels: The binary (0 or 1) labels for that training data Returns: The trained model """ # * START CODE HERE * # * END CODE HERE *

This function should fit a Naive Bayes model given a training matrix and labels. The function should return the state of that model. Feel free to use whatever datatype you wish for the state of the...

All of Statistics

6. Advanced Topics: This section covers advanced topics in statistics, including high-dimensional data analysis, time series analysis, and causal inference. Throughout the book, Wasserman emphasizes ...

def predict_from_naive_bayes_model(model, matrix): """Use a Naive Bayes model to compute predictions for a target matrix. This function should be able to predict on the models that fit_naive_bayes_model outputs. Args: model: A trained model from fit_naive_bayes_model matrix: A numpy array containing word counts Returns: A numpy array containg the predictions from the model """ # * START CODE HERE * log_probabilities = [] for i in range(2): log_probabilities.append(np.sum(matrix * model.log_class_priors[i] + model.log_likelihoods[i])) return np.argmax(log_probabilities) # Return the class with the highest probability # * END CODE HERE *代码报错AttributeError: 'dict' object has no attribute 'log_class_priors'请帮我修改

"""Use a Naive Bayes model to compute predictions for a target matrix. This function should be able to predict on the models that fit_naive_bayes_model outputs. Args: model: A trained model from ...

matlab classification learning

To use these methods, you will need to prepare your data and choose appropriate features for your classification model. Matlab provides tools for data preprocessing, feature selection, and model ...

inference in Bayes

We then update this prior distribution with the observed data using Bayes' theorem to obtain a posterior distribution that reflects our updated beliefs about the probability of events or parameters ...

请将以下代码：model1 <- train(data_train, y1_train, method="lm", trControl=ctrl) model2 <- train(data_train, y2_train, method="lm", trControl=ctrl) model3 <- train(data_train, y3_train, method="lm", trControl=ctrl)。参照以下代码改写：trainIndex <- createDataPartition(iris$Species,p=split,list=FALSE) data_train <- iris[ trainIndex,] data_test <- iris[-trainIndex,] model <- NaiveBayes(Species~.,data=data_train)

以下是参照你给出的代码改写后的代码： ``` trainIndex (iris$Species, p = split, list = FALSE) ...这个例子中使用的是 `train()` 函数，而不是 `NaiveBayes()` 函数，因为它是用于训练线性回归模型的。

import pandas as pd import numpy as np import matplotlib.pyplot as plt import sklearn.model_selection as sM_S import sklearn.preprocessing as sP import sklearn.tree as sT import sklearn.metrics as sM import sklearn.naive_bayes as sNB #1读取数据 data_train = pd.read_excel("train data.xlsx") # 1.1剔除无关列 data = data_train.iloc[:,3:-1] #2预处理 data = data.dropna() #3.数据预处理：空值处理，值映射（分段），归一化/标准化 X = data.iloc[:,0:-1] y = data.iloc[:,-1] mms = sP.MinMaxScaler() X = mms.fit_transform(X) #4.分割数据集和测试集 x_train,x_text,y_train,y_text = sM_S.train_test_split(X,y,test_size=0.33,random_state=42) #5.选择模型 model = sT.DecisionTreeClassifier(max_depth=7) #6.训练模型 model.fit(x_train,y_train) #7.评价模型，赛事要求用F1 y_predict = model.predict(x_text) score = sM.f1_score(y_predict,y_text,average="macro") print("预处理：{} 模型：{} 参数：{} 得分：{}".format("均值填充处理_归一化","决策树","max_depth=7",score)) # 8.应用模型--预测 trainData = pd.read_excel("train data.xlsx") trainData = pd.DataFrame(trainData) trainData = trainData.iloc[:, 3:-2] trainData = trainData.dropna() # 删除空值行 val_data = trainData val_data = mms.fit_transform(val_data) print("预测 train data.xlsx 的结果为：", model.predict(val_data)) 以上这段代码能运行吗？

import sklearn.naive_bayes as sNB # 1.读取数据 data_train = pd.read_excel("train data.xlsx") # 1.1剔除无关列 data = data_train.iloc[:, 3:-1] # 2.预处理 data = data.dropna() # 3.数据预处理：空值...

handle data

When handling data for NLP tasks, it is important to preprocess the data by cleaning and formatting it appropriately. This may involve removing any irrelevant information, tokenizing the text into ...

为什么以下代码读出的lab4_result.txt显示的结果乱码，请对以下代码进行修改import pandas as pd from sklearn.feature_extraction.text import TfidfVectorizer from sklearn.naive_bayes import MultinomialNB from sklearn.metrics import accuracy_score # 读取训练数据和测试数据 train_data = pd.read_csv('data_train.csv') test_data = pd.read_csv('data_test.csv') # 提取特征向量 vectorizer = TfidfVectorizer() X_train = vectorizer.fit_transform(train_data['title'] + ' ' + train_data['keywords']) X_test = vectorizer.transform(test_data['title'] + ' ' + test_data['keywords']) # 训练模型 model = MultinomialNB() model.fit(X_train, train_data['label']) # 预测结果 y_pred = model.predict(X_test) # 输出分类结果 with open('lab4_result.txt', 'w') as f: for i, y in enumerate(y_pred): f.write('学号姓名{}\n'.format(y))

from sklearn.naive_bayes import MultinomialNB from sklearn.metrics import accuracy_score # 读取训练数据和测试数据 train_data = pd.read_csv('data_train.csv', encoding='utf-8') test_data = pd.read_csv...

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