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Probability Theory_The Logic of Science

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Probability Theory_The Logic of Science "The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind." ---James Clerk Maxwell (1850)

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Probability Theory:

The Logic of Science

E. T. Jaynes

Wayman Crow Professor of Physics

Washington University

St. Louis, MO 63130, U. S. A.

Dedicated to the Memory of Sir Harold Jereys,

who saw the truth and preserved it.

Fragmentary Edition of March 1996. Copyright



1995 by Edwin T. Jaynes.

i i

PROBABILITY THEORY { THE LOGIC OF SCIENCE

Short Contents

PART A - PRINCIPLES AND ELEMENTARY APPLICATIONS

Chapter 1 Plausible Reasoning

Chapter 2 Quantitative Rules: The Cox Theorems

Chapter 3 Elementary Sampling Theory

Chapter 4 Elementary Hypothesis Testing

Chapter 5 Queer Uses for Probability Theory

Chapter 6 Elementary Parameter Estimation

Chapter 7 The Central Gaussian, or Normal, Distribution

Chapter 8 Suciency, Ancillarity, and All That

Chapter 9 Repetitive Exp eriments: Probability and Frequency

Chapter 10 Physics of \Random Experiments"

Chapter 11 The Entropy Principle

Chapter 12 Ignorance Priors { Transformation Groups

Chapter 13 Decision Theory: Historical Survey

Chapter 14 Simple Applications of Decision Theory

Chapter 15 Paradoxes of Probability Theory

Chapter 16 Orthodox Statistics: Historical Background

Chapter 17 Principles and Pathology of Orthodox Statistics

Chapter 18 The A

{Distribution and Rule of Succession

PART B{ADVANCED APPLICATIONS

Chapter 19 Physical Measurements

Chapter 20 Regression and Linear Mo dels

Chapter 21 Estimation with Cauchy and

{Distributions

Chapter 22 Time Series Analysis and Autoregressive Mo dels

Chapter 23 Spectrum / Shap e Analysis

Chapter 24 Model Comparison and Robustness

Chapter 25 Image Reconstruction

Chapter 26 Marginalization Theory

Chapter 27 Communication Theory

Chapter 28 Optimal Antenna and Filter Design

Chapter 29 Statistical Mechanics

Chapter 30 Maximum Entropy { Matrix Formulation

APPENDICES

Appendix A Other Approaches to Probability Theory

Appendix B Formalities and Mathematical Style

Appendix C Convolutions and Cumulants

Appendix D Dirichlet Integrals and Generating Functions

Appendix E The Binomial { Gaussian Hierarchy of Distributions

Appendix F Fourier Analysis

Appendix G Innite Series

Appendix H Matrix Analysis and Computation

Appendix I Computer Programs

REFERENCES

ii ii

PROBABILITY THEORY { THE LOGIC OF SCIENCE

Long Contents

PART A { PRINCIPLES and ELEMENTARY APPLICATIONS

Chapter 1 PLAUSIBLE REASONING

Deductive and Plausible Reasoning 101

Analogies with Physical Theories 103

The Thinking Computer 104

Introducing the Rob ot 105

Boolean Algebra 106

Adequate Sets of Operations 108

The Basic Desiderata 111

COMMENTS 114

Common Language vs. Formal Logic 115

Nitpicking 116

Chapter 2 THE QUANTITATIVE RULES

The Product Rule 201

The Sum Rule 206

Qualitative Properties 210

Numerical Values 212

Notation and Finite Sets Policy 217

COMMENTS 218

\Sub jective" vs. \Ob jective" 218

Godel's Theorem 218

Venn Diagrams 220

The \Kolmogorov Axioms" 222

Chapter 3 ELEMENTARY SAMPLING THEORY

Sampling Without Replacement 301

Logic Versus Propensity 308

Reasoning from Less Precise Information 311

Expectations 313

Other Forms and Extensions 314

Probability as a Mathematical Tool 315

The Binomial Distribution 315

Sampling With Replacement 318

Digression: A Sermon on Reality vs. Mo dels 318

Correction for Correlations 320

Simplication 326

COMMENTS 327

A Look Ahead 328

iii CONTENTS iii

Chapter 4 ELEMENTARY HYPOTHESIS TESTING

Prior Probabilities 401

Testing Binary Hypotheses with Binary Data 404

Non{Extensibility Beyond the Binary Case 410

Multiple Hypothesis Testing 411

Continuous Probability Distributions (pdf 's) 418

Testing an Innite Number of Hypotheses 420

Simple and Compound (or Comp osite) Hypotheses 424

COMMENTS 425

Etymology 425

What HaveWe Accomplished? 426

Chapter 5 QUEER USES FOR PROBABILITY THEORY

Extrasensory Perception 501

Mrs. Stewart's Telepathic Powers 502

Converging and Diverging Views 507

Visual Perception { Evolution into Bayesianity? 512

The Discovery of Neptune 513

Digression on Alternative Hypotheses 514

Horseracing and Weather Forecasting 518

Paradoxes of Intuition 521

Bayesian Jurisprudence 521

COMMENTS 523

Chapter 6 ELEMENTARYPARAMETER ESTIMATION

Inversion of the Urn Distributions 601

Both N and R Unknown 601

Uniform Prior 604

Truncated Uniform Priors 607

A Concave Prior 609

The Binomial Monkey Prior 610

Metamorphosis into Continuous Parameter Estimation 612

Estimation with a Binomial Sampling Distribution 613

Digression on Optional Stopping 615

The Likelihood Principle 616

Compound Estimation Problems 617

A Simple Bayesian Estimate: Quantitative Prior Information 618

From Posterior Distribution to Estimate 621

Back to the Problem 624

Eects of Qualitative Prior Information 626

The Jereys Prior 629

The PointofitAll 630

Interval Estimation 632

Calculation of Variance 632

Generalization and Asymptotic Forms 634

A More Careful Asymptotic Derivation 635

COMMENTS 636

iv CONTENTS iv

Chapter 7 THE CENTRAL GAUSSIAN, OR NORMAL DISTRIBUTION

The Gravitating Phenomenon 701

The Herschel{Maxwell Derivation 702

The Gauss Derivation 703

Historical Imp ortance of Gauss' Result 704

The Landon Derivation 705

Why the Ubiquitous Use of Gaussian Distributions? 707

Why the Ubiquitous Success? 709

The Near{Irrelevance of Sampling Distributions 711

The Remarkable Eciency of Information Transfer 712

Nuisance Parameters as Safety Devices 713

More General Properties 714

Convolution of Gaussians 715

Galton's Discovery 715

Population Dynamics and Darwinian Evolution 717

Resolution of Distributions into Gaussians 719

The Central Limit Theorem 722

Accuracy of Computations 723

COMMENTS 724

Terminology Again 724

The Great Inequality of Jupiter and Saturn 726

Chapter 8 SUFFICIENCY, ANCILLARITY, AND ALL THAT

Suciency 801

Fisher Suciency 803

Generalized Suciency 804

Examples

Suciency Plus Nuisance Parameters

The Pitman{Koopman Theorem

The Likelihood Principle

Eect of Nuisance Parameters

Use of Ancillary Information

Relation to the Likelihoo d Principle

Asymptotic Likelihood: Fisher Information

Combining Evidence from Dierent Sources: Meta{Analysis

Pooling the Data

Fine{Grained Prop ositions: Sam's Broken Thermometer

COMMENTS

The Fallacy of Sample Re{use

AFolk{Theorem

Eect of Prior Information

Clever Tricks and Gamesmanship

Chapter 9 REPETITIVE EXPERIMENTS { PROBABILITY AND FREQUENCY

Physical Experiments 901

The Poorly Informed Rob ot 902

Induction 905

Partition Function Algorithms 907

Relation to Generating Functions 911

Another Way of Lo oking AtIt 912

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Probability Theory The Logic of Science

Probability Theory The Logic of Science Probability Theory The Logic of Science

E.T. Jaynes__Probability Theory, The Logic Of Science

简介 · · · · · · 　本书将概率和统计推断融合在一起，用新的观点生动地描述了概率论在物理学、数学、经济学、化学和生物学等领域中的广泛应用，尤其是它阐述了贝叶斯理论的丰富应用，弥补了其他概率和统计...

Probability Theory The Logic of Science English version

Probability theory the logic of science-概率论沉思录1

probability theory: the logic of science

概率论是一种科学的逻辑。它是一种数学工具，用来研究事物发生的可能性和不确定性。概率论被广泛地应用于各个领域，如物理学、经济学、生物学等，它为科学研究提供了理论基础和方法论。概率论的逻辑在于它通过数学...

下面代码报错请修改一下：from pypinyin import pinyin import sys # 定义拼音汉字映射表，可以根据需要扩充 pinyin_dict = { 'a': '阿啊呵腌嗄锕', 'ai': '爱埃碍挨唉哎哀皑癌矮艾蔼隘碍捱嗳嫒霭', 'an': '安按暗岸案俺鞍氨庵胺谙埯揞犴', # ... } # 接收用户输入的拼音串 input_pinyin = input('请输入拼音串：') # 将拼音串转化为列表 pinyin_list = input_pinyin.split(' ') # 定义一个函数，用于从映射表中查找可能性最大的汉字 def find_max_probability_char(pinyin_char, candidate_chars): max_probability = 0 max_probability_char = '' for char in candidate_chars: probability = pinyin.get(pinyin_char, format='numerical')[0][0] if probability > max_probability: max_probability = probability max_probability_char = char return max_probability_char # 遍历每个拼音，查找可能性最大的汉字 result = '' for pinyin_char in pinyin_list: if pinyin_char in pinyin_dict: candidate_chars = pinyin_dict[pinyin_char] max_probability_char = find_max_probability_char(pinyin_char, candidate_chars) result += max_probability_char else: result += pinyin_char # 输出结果 print(result)

在代码中，pinyin.get应该修改... max_probability_char = find_max_probability_char(pinyin_char, candidate_chars) result += max_probability_char else: result += pinyin_char # 输出结果 print(result) ```

用贝叶斯公式说明当出现大力字样的时候，80%概率发生上涨的概率，python代码

def probability_of_up_given_dalao(probability_of_up, probability_of_dalao_given_up): probability_of_dalao = 0.5 return probability_of_dalao_given_up * probability_of_up / probability_of_dalao ...

Please write a genetic algorithm in Python to calculate the maximum value of function f (x)=0.4+sinc (4x)+1.1sinc (4x+2)+0.8sinc (X-2)+0.7sinc (6x-4) between -2 and 2. The function solution is represented by a 16 bit binary number. The population size is 30, the crossover probability is 0.3, the mutation probability is 0.01, and the number of iterations is 400

This is the maximum value of the `f` function that the genetic algorithm was able to find in 400 iterations with a population size of 30, a crossover probability of 0.3, and a mutation probability of ...

帮我补充代码：class LanguageModel(LanguageModel): def get_unigram_logp(self, unigram): """Computes the log-probability of unigram under this LanguageModel. Args: unigram (str): Unigram for which to compute the log-probability. Returns: log_p (float): Log-probability of unigram under this LanguageModel. """ ### Begin your code ### End your code def get_bigram_logp(self, w_1, w_2): """Computes the log-probability of unigram under this LanguageModel. Note: Use self.lambda_ for the unigram-bigram interpolation factor. Args: w_1 (str): First word in bigram. w_2 (str): Second word in bigram. Returns: log_p (float): Log-probability of bigram under this LanguageModel. """ ### Begin your code ### End your code def get_query_logp(self, query): """Computes the log-probability of query under this LanguageModel. Args: query (str): Whitespace-delimited sequence of terms in the query. Returns: log_p (float): Log-probability assigned to the query under this LanguageModel. """ ### Begin your code ### End your code

"""Computes the log-probability of unigram under this LanguageModel. Args: unigram (str): Unigram for which to compute the log-probability. Returns: log_p (float): Log-probability of unigram ...

counting process theory

Counting process theory is a branch of probability theory that deals with stochastic processes that count the occurrence of certain events over time. It is also known as point process theory or ...

You could also create code to simulate the Monty Hall problem. Create a function named random_door, which uses numpy.random.choice to sample 1 door randomly from a list of integer door indices (1-3 in this case). Use this function to randomly select the door the car is behind and the contestant’s initial choice of doors. Create a function monty_choice, which chooses the door Monty opens, conditional on the contestant’s choice of doors and the door with the car. For the case where the contestant has selected the door with the car, select the door to open by simulating the flip of a fair coin using the np.random.binomial function with n = 1. Create a function win_car, which determines if the contestant wins the car, conditional on the strategy selected, {switch, noswitch}, the door the contestant selected, the door with the car, and the door Monty opened. Create a function named simulation that allows you to run the simulation n = 1000 times, with a switch or noswitch strategy. Execute your simulation for each possible strategy. For the two strategies, plot side by side bar charts showing the numbers of successes and failures for each strategy. Describe the strategy a contestant should adopt for this game. How much will the chosen strategy change the probability of winning a car? Is this result consistent with the conditional probability of this problem?

This result is consistent with the conditional probability of the problem, which shows that the probability of winning the car is 2/3 if the contestant switches doors, and 1/3 if they do not switch.

import numpy as np def loaddata(): X = np.array([[1,'S'],[1,'M'],[1,'M'],[1,'S'], [1, 'S'], [2, 'S'], [2, 'M'], [2, 'M'], [2, 'L'], [2, 'L'], [3, 'L'], [3, 'M'], [3, 'M'], [3, 'L'], [3, 'L']]) y = np.array([-1,-1,1,1,-1,-1,-1,1,1,1,1,1,1,1,-1]) return X, y def Train(trainset,train_labels): m = trainset.shape[0] n = trainset.shape[1] prior_probability = {}# 先验概率 key是类别值，value是类别的概率值 conditional_probability ={}# 条件概率 key的构造：类别，特征,特征值 #类别的可能取值 labels = set(train_labels) # 计算先验概率(此时没有除以总数据量m) for label in labels: prior_probability[label] = len(train_labels[train_labels == label])+1 #计算条件概率 for i in range(m): for j in range(n): # key的构造：类别，特征,特征值 #补充计算条件概率的代码-1； key = str(train_labels[i])+','+str(j)+','+str(trainset[i][j]) conditional_probability[key] = (conditional_probability[key]+1 if (key in conditional_probability) else 1) conditional_probability_final = {} for key in conditional_probability: #补充计算条件概率的代码-2； label = key.split(',')[0] conditional_probability[key]+=1 key1 = int(key.split(',')[1]) Ni = len(set(trainset[:,key1])) conditional_probability_final[key] =conditional_probability[key]/(prior_probability[int(label)]+Ni) # 最终的先验概率(此时除以总数据量m) for label in labels: prior_probability[label] = prior_probability[label]/ (m+len(labels)) return prior_probability,conditional_probability_final,labels def predict(data): result={} for label in train_labels_set: temp=1.0 #补充预测代码； print('result=',result) #排序返回标签值 result[label] = tempprior_probability[label] for i in range (len(data)): key = str(label)+ ','+str(i)+','+str(data[i]) result[label]=conditional_probability_final[key] print('result=',result) #排序返回标签值 return sorted(result.items(), key=lambda x: x[1],reverse=True)[0][0] X,y = loaddata() prior_probability,conditional_probability,train_labels_set = Train(X,y) r_label = predict([2,'S']) print(' r_label =', r_label)运行次python代码

其中loaddata()函数载入训练数据集X和标签集y，Train()函数训练得到先验概率prior_probability和条件概率conditional_probability_final，predict()函数根据输入的测试数据进行分类预测。具体来说，Train()函数中...

Please write a genetic algorithm in Python to calculate the maximum value of function f (x)=0.4+sinc (4x)+1.1sinc (4x+2)+0.8sinc (x-2)+0.7sinc (6x-4) between -2 and 2, and the value of x when the maximum value is taken. The function solution is expressed in 16 bit binary numbers, with population size of 30, crossover probability of 0.3, mutation probability of 0.01, and iteration times of 400

It takes as input the population size, crossover probability, mutation probability, and number of iterations. It returns the best individual and its fitness value. - The algorithm initializes a ...

Optimize the following code to use the variable: variance in the code. def e_step(xs: np.ndarray, ys: np.ndarray, affine: np.ndarray, translation: np.ndarray, variance: float) -> np.ndarray: """ The e-step of the em algorithm, estimating the responsibility P=[p(y_m | x_n)] based on current model :param xs: a set of points with size (N, D), N is the number of samples, D is the dimension of points :param ys: a set of points with size (M, D), M is the number of samples, D is the dimension of points :param affine: an affine matrix with size (D, D) :param translation: a translation vector with size (1, D) :param variance: a float controlling the variance of each Gaussian component :return: the responsibility matrix P=[p(y_m | x_n)] with size (N, M), which row is the conditional probability of clusters given the n-th sample x_n """ # TODO: Change the code below and implement the E-step of GMM responsibility = np.ones((xs.shape[0], ys.shape[0])) / ys.shape[0] for n in range(xs.shape[0]): for m in range(ys.shape[0]): temp = -0.5 * np.linalg.norm(xs[n] - ys[m] @ affine - translation) ** 2 responsibility[n, m] = 1 / (2 * np.pi) ** (xs.shape[1] / 2) * np.exp(temp) return responsibility / np.sum(responsibility, axis=1, keepdims=True)

which row is the conditional probability of clusters given the n-th sample x_n """ # Calculate the constant term in the Gaussian distribution constant_term = 1 / (2 * np.pi * variance) ** (xs....

在mysql analytic_double表中，增加字段名称为从1_probability 到44_probability的类型为 varchar(255) COLLATE utf8_bin DEFAULT NULL字段语句应该怎么写

要在 MySQL 的 analytic_double 表中增加从 1_probability 到 44_probability 的字段，类型为 varchar(255) COLLATE utf8_bin DEFAULT NULL，可以使用 ALTER TABLE 语句进行操作。以下是具体的 SQL 语句示例： ``` ...

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Probability Theory_The Logic of Science

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Probability Theory_The Logic of Science

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Probability Theory The Logic of Science

E.T. Jaynes__Probability Theory, The Logic Of Science

Probability Theory The Logic of Science English version

Probability theory the logic of science-概率论沉思录1

probability theory: the logic of science

用贝叶斯公式说明当出现大力字样的时候，80%概率发生上涨的概率，python代码

counting process theory

写一段matlab计算图像信息熵

probability theory and example

prediction interval coverage probability

在mysql analytic_double表中，增加字段名称为 从1_probability 到44_probability的字段 应该怎么写

Excess probability

在mysql analytic_double表中，增加字段名称为 从1_probability 到44_probability的类型为 varchar(255) COLLATE utf8_bin DEFAULT NULL字段 语句应该怎么写

会员权益专享

最新资源

在mysql analytic_double表中，增加字段名称为从1_probability 到44_probability的字段应该怎么写

在mysql analytic_double表中，增加字段名称为从1_probability 到44_probability的类型为 varchar(255) COLLATE utf8_bin DEFAULT NULL字段语句应该怎么写