Bayesian Inference : Parameter Estimation and Decisions

"The book provides a generalization of Gaussian error intervals to situations where the data follow non-Gaussian distributions. This usually occurs in frontier science, where the observed parameter is just above background or the histogram of multiparametric data contains empty bins. Then the validity of a theory cannot be decided by the chi-squared-criterion, but this long-standing problem is solved here. The book is based on Bayes' theorem, symmetry and differential geometry. In addition to solutions of practical problems, the text provides an epistemic insight. The logic of quantum mechanics is obtained as the logic of unbiased inference from counting data. However, no knowledge of quantum mechanics is required. The text examples and exercises are written at an introductory level."--BOOK JACKET Filling a longstanding need in the physical sciences, Bayesian Inference offers the first basic introduction for advanced undergraduates and graduates in the physical sciences. This text and reference generalizes Gaussian error intervals to situations in which the data follow distributions other than Gaussian. This usually occurs in frontier science because the observed parameter is barely above the background or the histogram of multiparametric data contains many empty bins. In this case, the determination of the validity of a theory cannot be based on the chi-squared-criterion. In addition to the solutions of practical problems, this approach provides an epistemic insight: the logic of quantum mechanics is obtained as the logic of unbiased inference from counting data. Requiring no knowledge of quantum mechanics, the text is written on introductory level, with many examples and exercises, for physicists planning to, or working in, fields such as medical physics, nuclear physics, quantum mechanics, and chaos. Front Matter....Pages I-XIII Knowledge and Logic....Pages 1-7 Bayes’ Theorem....Pages 8-18 Probable and Improbable Data....Pages 19-28 Description of Distributions I: Real x ....Pages 29-39 Description of Distributions II: Natural x ....Pages 40-45 Form Invariance I: Real x ....Pages 46-56 Examples of Invariant Measures....Pages 57-64 A Linear Representation of Form Invariance....Pages 65-70 Beyond Form Invariance: The Geometric Prior....Pages 71-80 Inferring the Mean or Standard Deviation....Pages 81-94 Form Invariance II: Natural x ....Pages 95-108 Independence of Parameters....Pages 109-119 The Art of Fitting I: Real x ....Pages 120-129 Judging a Fit I: Real x ....Pages 130-136 The Art of Fitting II: Natural x ....Pages 137-152 Judging a Fit II: Natural x ....Pages 153-161 Summary....Pages 162-167 Back Matter....Pages 169-265 Knowledge and Logic Bayes' Theorem Probable and improbable Data Bayes' Theorem and the Truth Description of Distributions I: Probability Densities Description of Distributions II: Form Invariance I: Real x Examples of Invariant Measures A Linear Representation of Form Invariance Beyond Form Invariance: The Geometric Prior Econophysics Inferring Mean or Standard Deviation Form Invariance II: Natural x Independence of Parameters The Art of Fitting I: Real x Judging a Fit I: Real x The Art of Fitting II: Natural x Judging a Fit II: Natural x Summary A Problems B Form Invariance I: Probability Densities C Beyond Form Invariance: The Geometric Prior D Inferring Mean or Standard Deviation E Form Invariance II: Natural x F Independence of Parameters G The Art of Fitting I: Real x H Judging a Fit II: Natural x Bibliography. . Solving a longstanding problem in the physical sciences, this text and reference generalizes Gaussian error intervals to situations in which the data follow distributions other than Gaussian. The text is written at introductory level, with many examples and exercises.