Bayesian Inference

1 Knowledge and Logic.- 2 Bayes’ Theorem.- 3 Probable and Improbable Data.- 4 Description of Distributions I: Real x.- 5 Description of Distributions II: Natural x.- 6 Form Invariance I: Real x.- 7 Examples of Invariant Measures.- 8 A Linear Representation of Form Invariance.- 9 Beyond Form Invariance: The Geometric Prior.- 10 Inferring the Mean or Standard Deviation.- 11 Form Invariance II: Natural x.- 12 Independence of Parameters.- 13 The Art of Fitting I: Real x.- 14 Judging a Fit I: Real x.- 15 The Art of Fitting II: Natural x.- 16 Judging a Fit II: Natural x.- 17 Summary.- A Problems and Solutions.- A.1 Knowledge and Logic.- A.2 Bayes’ Theorem.- A.3 Probable and Improbable Data.- A.7 Examples of Invariant Measures.- A.8 A Linear Representation of Form Invariance.- A.9 Beyond Form Invariance: The Geometric Prior.- A.10 Inferring the Mean or Standard Deviation.- A.12 Independence of Parameters.- B.1 The Correlation Matrix.- B.2 Calculation of a Jacobian.- B.4 The Beta Function.- C.1 The Multinomial Theorem.- D Form Invariance I: Probability Densities.- D.1 The Invariant Measure of a Group.- E Beyond Form Invariance: The Geometric Prior.- E.1 The Definition of the Fisher Matrix.- E.2 Evaluation of a Determinant.- E.3 Evaluation of a Fisher Matrix.- E.4 The Fisher Matrix of the Multinomial Model.- F Inferring the Mean or Standard Deviation.- G.1 Destruction and Creation Operators.- G.2 Unitary Operators.- G.3 The Probability Amplitude of the Histogram.- G.4 Form Invariance of the Histogram.- G.5 Quasi-Events in the Histogram.- G.6 Form Invariance of the Binomial Model.- G.7 Conservation of the Number of Events.- G.8 Normalising the Posterior of the Binomial Model.- G.9 Lack of Form Invariance of the Multinomial Model.- H Independence of Parameters.- H.1 On the Measure of a Factorising Group.- H.2 Marginal Distribution of the Posterior of the Multinomial Model.- H.3 A Minor Posterior of the Multinomial Model.- I.1 A Factorising Gaussian Model.- I.2 A Basis for Fourier Expansions.- J.2 The Deviation Between Two Distributions.- References.