Practical Bayesian Inference. A Primer for Physical Scientists

Description

Preface......Page 2 The three doors problem......Page 5 Probability......Page 7 Expectation, variance, and moments......Page 11 Univariate probability distributions......Page 14 Cumulative distribution function and quantile function......Page 24 Multiple variables......Page 25 Combinations and permutations......Page 29 Random number generation......Page 32 Change of variables......Page 34 The three doors problem revisited......Page 36 Estimators......Page 37 Noise, measurement errors, and measurement models......Page 38 reduction......Page 40 Population properties vs sample estimates......Page 45 The mean is not necessarily the most efficient estimator......Page 46 Outliers and robust statistics......Page 48 Errors, accuracy, and precision......Page 50 Propagation and combination of uncertainties......Page 51 Introduction to data modelling......Page 56 Bayesian model comparison......Page 57 Data modelling with parametric models......Page 62 Making decisions......Page 67 Estimating one parameter from one data point......Page 68 An inference story......Page 75 One-dimensional linear regression......Page 77 Linear models......Page 84 Maximum likelihood and maximum posterior probability......Page 85 Multi-dimensional linear regression......Page 90 One-dimensional regression with nonlinear functions......Page 92 Least squares with errors on both axes......Page 95 Bias-variance decomposition......Page 96 Bayesian analysis of coin tossing......Page 99 Likelihoods can be arbitrarily small and their absolute values are irrelevant......Page 110 Assigning priors......Page 113 Some other conjugate priors......Page 121 Summarizing distributions......Page 122 Conditional and marginal distributions......Page 126 Inferring the parameters of a Gaussian......Page 127 A two-parameter problem: estimating amplitude and background......Page 134 The quadratic approximation......Page 143 Density estimation......Page 146 Why we need efficient sampling......Page 155 Uses of integration in Bayesian inference......Page 157 Monte Carlo integration......Page 158 Monte Carlo sampling......Page 160 Markov Chain Monte Carlo......Page 162 R code......Page 168 Fitting a straight line with unknown noise......Page 174 Fitting a quadratic curve with unknown noise......Page 191 A mixture model: fitting a straight line with an outlier model......Page 194 Fitting curves with arbitrary error bars on both axes......Page 200 Fitting a straight line with unknown noise......Page 205 Fitting a quadratic curve with unknown noise......Page 222 A mixture model: fitting a straight line with an outlier model......Page 225 Fitting curves with arbitrary error bars on both axes......Page 231 The principles of frequentist statistics and hypothesis testing......Page 236 One-sample hypothesis tests......Page 238 Two-sample hypothesis tests......Page 247 Hypothesis testing in linear modelling......Page 249 distribution......Page 250 Issues with frequentist hypothesis testing......Page 254 Bayesian model comparison......Page 257 Example of an analytic evidence calculation: is a coin fair?......Page 259 Example of a numerical evidence calculation: is there evidence for a non-zero gradient?......Page 263 Comparing Gaussians (or other distributions)......Page 270 How the evidence accounts for model complexity......Page 271 Other ways of computing the evidence......Page 274 Other measures for model comparison: AIC and BIC......Page 277 The stopping problem......Page 280 Issues with Bayesian model comparison......Page 284 Cross-validation......Page 287 Regularization in regression......Page 292 Regression with basis functions......Page 298 Regression kernels......Page 305 A non-parametric smoothing problem......Page 309 Numerical optimization (mode finding)......Page 312 Bootstrap resampling......Page 314 Refs......Page 320