Bayesian Field Theory

Ask a traditional mathematician the likely outcome of a coin-toss, and he will reply that no evidence exists on which to base such a prediction. Ask a Bayesian, and he will examine the coin, conclude that it was probably not tampered with, and predict 500 heads in 1000 tosses; a subsequent experiment would then be used to refine this prediction. The Bayesian approach, in other words, permits the use of prior knowledge when testing a hypothesis. Long the province of mathematicians and statisticians, Bayesian methods are applied in this book to problems in cutting-edge physics. Jorg Lemm offers practical examples of Bayesian analysis for the physicist working in such areas as neural networks, artificial intelligence and inverse problems in quantum theory. The book also includes non-parametric density estimation problems, including, as special cases, non-parametric regression and pattern recognition. The thought-provoking text should be of interest to physicists as well as to other specialists in the rapidly growing number of fields that make use of Bayesian methods.

Table of Contents:
List of Figures List of Tables List of Numerical Case Studies Acknowledgments Part I: Introduction Part II: Bayesian Framework Chapter 1. Bayesian Models Chapter 2. Bayesian Decision Theory Chapter 3. A Priori Information Part III: Gaussian Prior Factors Chapter 4. Gaussian Prior Factor for Log-Likelihoods Chapter 5. Gaussian Prior Factor For Likelihoods Chapter 6. Quadratic Density Estimation and Empirical Risk Minimization Chapter 7. Numerical Case Study: Density Estimation with Gaussian Specific Priors Chapter 8. Gaussian Prior Factors for General Fields Chapter 9. Covariances and Invariances Chapter 10. Non-Zero Means Chapter 11. Regression Chapter 12. Classification Part IV: Parameterizing Likelihoods: Variational Methods Chapter 13. General Likelihood Parameterizations Chapter 14. Gaussian Priors for Likelihood Parameterizations Chapter 15. Linear Trial Spaces Chapter 16. Linear Regression Chapter 17. Mixture Models Chapter 18. Additive Models Chapter 19. Product Ansatz Chapter 20. Decision Trees Chapter 21. Projection Pursuit Chapter 22. Neural Networks Part V: Parameterizing Priors: Hyperparameters Chapter 23. Quenched and Annealed Prior Normalization Chapter 24. Saddle Point Approximations and Hyperparameters Chapter 25. Adapting Prior Means Chapter 26. Adapting Prior Covariances Chapter 27. Integer Hyperparameters Chapter 28. Hyperfields Chapter 29. Auxiliary Fields Chapter 30. Non-Quadratic Potentials Part VI: Mixtures of Gaussian Prior Factors Chapter 31. Multimodal Energy Surfaces Chapter 32. Prior Mixtures for Density Estimation Chapter 33. Numerical Case Study: Prior Mixtures for Density Estimation Chapter 34. Prior Mixtures for Regression Chapter 35. Local Mixtures Chapter 36. Numerical Case Study: Image Completion Part VII: Bayesian Inverse Quantum Theory (BIQT) Chapter 37. Bayesian Inverse Quantum Statistics (BIQS) Chapter 38. Bayesian Inverse Time-Dependent Quantum Theory (BITDQ) Chapter 39. Bayesian Inverse Many-Body Theory Part VIII: Summary Bibliography Index

About the Author :
Jorg C. Lemm is a former teacher of physics and psychology at the University of Muenster, Germany, and has worked in the areas of statistics, decision theory, and neural networks.

Review :
There is a considerable amount of interesting discussion on inference generally and, in particular, on Bayesian inference. While a statistician might find the language and point-of-view somewhat different, this is a useful resource for those curious about the use of statistics in modern physics. -- Michael J. Evans Mathematical Reviews Thought-provoking and challenging. -- David A. Stephens Journal of the American Statistical Association The value of a statistical method outlined by [the Rev. Thomas] Bayes in 1763 has become increasingly apparent and has resulted in a blossoming of 'Bayesian' methods in scientific fields ranging from archeology to computing. The Economist