Scientific Reasoning The Bayesian Approach

by Howson, Colin; Urbach, Peter

• ISBN13: 9780812695786
• ISBN10: 081269578X
• Edition: 3rd
• Format: Paperback
• Copyright: 2006-03-16
• Publisher: Open Court

Summary

"Scientific Reasoning: The Bayesian Approach explains, in an accessible style, those elements of the probability calculus that are relevant to Bayesian methods, and argues that the probability calculus is best regarded as a species of logic." "Howson and Urbach contrast the Bayesian with the 'classical' view that was so influential in the last century, and demonstrate that familiar classical procedures for evaluating statistical hypotheses, such as significance tests, point estimation, confidence intervals, and other techniques, provide an utterly false basis for scientific inference. They also expose the well-known non-probabilistic philosophies of Popper, Lakatos, and Kuhn as similarly unscientific." "Scientific Reasoning shows how Bayesian theory, by contrast with these increasingly discredited approaches, provides a unified and highly satisfactory account of scientific method, an account which practicing scientists and all those interested in the sciences ought to master."--BOOK JACKET.

Author Biography

Colin Howson is Reader in Philosophy at the London School of Economics. Peter Urbach is Reader in Philosophy at the London School of Economics, and author of Francis Bacon's Philosophy of Science.

Table of Contents

Preface to the Second Edition	p. xvii
Bayesian Principles	p. 1
Introduction	p. 3
The Problem of Induction	p. 3
Popper's Attempt to Solve the Problem of Induction	p. 4
Scientific Method in Practice	p. 6
Probabilistic Induction: The Bayesian Approach	p. 9
The Objectivity Ideal	p. 11
The Plan of the Book	p. 12
Exercises	p. 13
The Probability Calculus	p. 17
Introduction	p. 17
Some Logical Preliminaries	p. 18
The Probability Calculus	p. 20
The Axioms	p. 20
Two Different Interpretations of the Axioms	p. 22
Useful Theorems of the Calculus	p. 24
Random Variables	p. 29
Kolmogorov's Axioms	p. 30
Propositions	p. 31
Infinitary Operations	p. 33
Countable Additivity	p. 34
Exercises	p. 34
Distributions and Densities	p. 37
Distributions	p. 37
Probability Densities	p. 37
Expected Values	p. 39
The Mean and Standard Deviation	p. 39
Probabilistic Independence	p. 41
Conditional Distributions	p. 43
The Bivariate Normal	p. 44
The Binomial Distribution	p. 46
The Weak Law of Large Numbers	p. 47
Exercises	p. 49
The Classical and Logical Theories	p. 51
Introduction	p. 51
The Classical Theory	p. 51
The Principle of Indifference	p. 52
The Rule of Succession	p. 55
The Principle of Indifference and the Paradoxes	p. 59
Carnap's Logical Probability Measures	p. 62
Carnap's c[dagger] and c*	p. 62
The Dependence on A Priori Assumptions	p. 66
Exercises	p. 72
Subjective Probability	p. 75
Degrees of Belief and the Probability Calculus	p. 75
Betting Quotients and Degrees of Belief	p. 75
Why Should Degrees of Belief Obey the Probability Calculus?	p. 78
The Ramsey--de Finetti Theorem	p. 79
Conditional Betting-Quotients	p. 81
Fair Odds and Zero Expectations	p. 84
Fairness and Consistency	p. 85
Upper and Lower Probabilities	p. 87
Other Arguments for the Probability Calculus	p. 89
The Standard Dutch Book Argument	p. 89
Scoring Rules	p. 91
Using a Standard	p. 93
The Cox-Good-Lucas Argument	p. 93
Introducing Utilities	p. 94
Conclusion	p. 95
Exercises	p. 96
Updating Belief	p. 99
Bayesian Conditionalisation	p. 99
Jeffrey Conditionalisation	p. 105
Generalising Jeffrey's Rule to Partitions	p. 108
Dutch Books Again	p. 109
The Principle of Minimum Information	p. 110
Conclusion	p. 112
Exercises	p. 113
Bayesian Induction: Deterministic Theories	p. 115
Bayesian Versus Non-Bayesian Approaches	p. 117
The Bayesian Notion of Confirmation	p. 117
The Application of Bayes's Theorem	p. 118
Falsifying Hypotheses	p. 119
Checking a Consequence	p. 119
The Probability of the Evidence	p. 123
The Ravens Paradox	p. 126
The Design of Experiments	p. 130
The Duhem Problem	p. 131
The Problem	p. 131
Lakatos and Kuhn on the Duhem Problem	p. 134
The Duhem Problem Solved by Bayesian Means	p. 136
Good Data, Bad Data, and Data Too Good to Be True	p. 142
Ad Hoc Hypotheses	p. 146
Some Examples of Ad Hoc Hypotheses	p. 147
A Standard Account of Adhocness	p. 149
Popper's Defence of the Adhocness Criterion	p. 151
Why the Standard Account Must Be Wrong	p. 154
The Bayesian View of Ad Hoc Theories	p. 157
The Notion of Independent Evidence	p. 158
Infinitely Many Theories Compatible with the Data	p. 161
The Problem	p. 161
The Bayesian Approach to the Problem	p. 162
Conclusion	p. 164
Exercises	p. 164
Classical Inference in Statistics	p. 169
Fisher's Theory	p. 171
Falsificationism in Statistics	p. 171
Fisher's Theory	p. 174
Has Fisher's Theory a Rational Foundation?	p. 178
Which Test-Statistic?	p. 181
The Chi-Square Test	p. 183
Sufficient Statistics	p. 188
Conclusion	p. 192
The Neyman-Pearson Theory of Significance Tests	p. 193
An Outline of the Theory	p. 193
How the Neyman-Pearson Theory Improves on Fisher's	p. 198
The Choice of Critical Region	p. 199
The Choice of Test-Statistic and the Use of Sufficient Statistics	p. 200
Some Problems for the Neyman-Pearson Theory	p. 201
What Does It Mean to Accept and Reject a Hypothesis?	p. 201
The Neyman-Pearson Theory as an Account of Inductive Support	p. 204
A Well-Supported Hypothesis Rejected in a Significance Test	p. 206
A Subjective Element in Neyman-Pearson Testing: The Choice of Null Hypothesis	p. 207
A Further Subjective Element: Determining the Outcome Space	p. 208
Justifying the Stopping Rule	p. 211
Testing Composite Hypotheses	p. 213
Conclusion	p. 217
Exercises	p. 218
The Classical Theory of Estimation	p. 223
Introduction	p. 223
Point Estimation	p. 225
Sufficient Estimators	p. 226
Unbiased Estimators	p. 228
Consistent Estimators	p. 231
Efficient Estimators	p. 233
Interval Estimation	p. 235
Confidence Intervals	p. 235
The Categorical-Assertion Interpretation of Confidence Intervals	p. 237
The Subjective-Confidence Interpretation of Confidence Intervals	p. 239
The Stopping Rule Problem, Again	p. 241
Prior Knowledge	p. 243
The Multiplicity of Competing Intervals	p. 244
Principles of Sampling	p. 247
Random Sampling	p. 247
Judgment Sampling	p. 247
Objections to Judgment Sampling	p. 248
Some Advantages of Judgment Sampling	p. 250
Conclusion	p. 252
Exercises	p. 253
Statistical Inference in Practice	p. 255
Causal Hypotheses: Clinical and Agricultural Trials	p. 257
Introduction: The Problem	p. 257
Control and Randomization	p. 259
Significance-Test Justifications for Randomization	p. 262
The Problem of the Reference Population	p. 262
Fisher's Argument	p. 265
Some Difficulties with Fisher's Argument	p. 266
A Plausible Defence	p. 268
Why the Plausible Defence Doesn't Work	p. 270
The Eliminative-Induction Defence of Randomization	p. 271
Sequential Clinical Trials	p. 274
Practical and Ethical Considerations	p. 279
Conclusion	p. 281
Regression Analysis	p. 283
Introduction	p. 283
Simple Linear Regression	p. 283
The Method of Least Squares	p. 286
Why Least Squares?	p. 287
Intuition as a Justification	p. 287
The Gauss-Markov Justification	p. 291
The Maximum-Likelihood Justification	p. 293
Summary	p. 294
Prediction	p. 295
Prediction Intervals	p. 295
Prediction by Confidence Intervals	p. 296
Making a Further Prediction	p. 297
Examining the Form of a Regression	p. 298
Prior Knowledge	p. 299
Data Analysis	p. 302
Inspecting Scatter Plots	p. 303
Outliers	p. 304
Influential Points	p. 309
Conclusion	p. 313
Exercises	p. 314
The Bayesian Approach to Statistical Inference	p. 317
Objective Probability	p. 319
Introduction	p. 319
Von Mises's Frequency Theory	p. 320
Relative Frequencies in Collectives	p. 320
Probabilities in Collectives	p. 325
Independence in Derived Collectives	p. 328
Summary of the Main Features of Von Mises's Theory	p. 330
The Empirical Adequacy of Von Mises's Theory	p. 331
The Fast-Convergence Argument	p. 332
The Laws of Large Numbers Argument	p. 332
The Limits-Occur-Elsewhere-in-Science Argument	p. 334
Preliminary Conclusion	p. 337
Popper's Propensity Theory, and Single-Case Probabilities	p. 338
Popper's Propensity Theory	p. 338
Jacta Alea Est	p. 339
The Theory of Objective Chance	p. 342
A Bayesian Reconstruction of Von Mises's Theory	p. 344
Are Objective Probabilities Redundant?	p. 347
Exchangeability and the Existence of Objective Probability	p. 349
Conclusion	p. 351
Exercises	p. 351
Bayesian Induction: Statistical Hypotheses	p. 353
The Prior Distribution and the Question of Subjectivity	p. 353
Estimating the Mean of a Normal Population	p. 354
Estimating a Binomial Proportion	p. 357
Credible Intervals and Confidence Intervals	p. 359
The Principle of Stable Estimation	p. 361
Describing the Evidence	p. 363
Sufficient Statistics	p. 367
Methods of Sampling	p. 369
Testing Causal Hypotheses	p. 372
A Bayesian Analysis of Clinical Trials	p. 374
Clinical Trials without Randomization	p. 378
Conclusion	p. 381
Exercises	p. 382
Finale	p. 387
The Objections to the Subjective Bayesian Theory	p. 389
Introduction	p. 389
The Bayesian Theory Is Prejudiced in Favour of Weak Hypotheses	p. 389
The Prior Probability of Universal Hypotheses Must Be Zero	p. 391
Probabilistic Induction Is Impossible	p. 395
The Principal Principle Is Inconsistent (Miller's Paradox)	p. 398
The Paradox of Ideal Evidence	p. 401
Hypotheses Cannot Be Supported by Evidence Already Known	p. 403
P(h	p. 403
Evidence Doesn't Confirm Theories Constructed to Explain It	p. 408
The Principle of Explanatory Surplus	p. 409
Prediction Scores Higher Than Accommodation	p. 411
The Problem of Subjectivism	p. 413
Entropy, Symmetry, and Objectivity	p. 413
Simplicity	p. 417
People Are Not Bayesians	p. 420
The Dempster-Shafer Theory	p. 423
Belief Functions	p. 423
What Are Belief Functions?	p. 426
Representing Ignorance	p. 429
Evaluating Probabilities with Imprecise Information	p. 430
Are We Calibrated?	p. 432
Reliable Inductive Methods	p. 434
Finale	p. 437
Exercises	p. 439
Bibliography	p. 443
Index	p. 459
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