Sivia D.S., Skilling J. Data Analysis: A Bayesian Tutorial

Oxford, Oxford University Press, 2006. - 259 p.The monograph is devoted to the development of statistical analysis methods based on the Bayesian approach. The issues of constructing Bayesian estimates, testing hypotheses, choosing a priori distributions, nonparametric estimation, experiment planning, etc. are considered. Designed for researchers, graduate students and students in the field of mathematical statistics.The basics.
Introduction: deductive logic versus plausible reasoning, Probability: Cox and the rules for consistent reasoning,
Corollaries: Bayes’ theorem and marginalization, Some history: Bayes, Laplace and orthodox statistics, Outline of book.
Parameter estimation I.
Example 1: is this a fair coin?, Reliabilities: best estimates, error-bars and confidence intervals.
Example 2: Gaussian noise and averages.
Example 3: the lighthouse problem.
Parameter estimation II.
Example 4: amplitude of a signal in the presence of background.
Reliabilities: best estimates, correlations and error-bars.
Example 5: Gaussian noise revisited.
Algorithms: a numerical interlude, Approximations: maximum likelihood and least-squares, Error-propagation: changing variables.
Model selection.
Introduction: the story of Mr A and Mr B.
Example 6: how many lines are there?
Other examples: means, variance, dating and so on.
Assigning probabilities.
Ignorance: indifference and transformation groups, Testable information: the principle of maximum entropy,
MaxEnt examples: some common pdfs, Approximations: interconnections and simplifications, Hangups: priors versus likelihoods.
Non-parametric estimation.
Introduction: free-form solutions, MaxEnt: images, monkeys and a non-uniform prior,
Smoothness: fuzzy pixels and spatial correlations, Generalizations: some extensions and comments.
Experimental design.
Introduction: general issues.
Example 7: optimizing resolution functions.
Calibration, model selection and binning, Information gain: quantifying the worth of an experiment.
Least-squares extensions.
Introduction: constraints and restraints, Noise scaling: a simple global adjustment, Outliers: dealing with erratic data,
Background removal, Correlated noise: avoiding over-counting, Log-normal: least-squares for magnitude data.
Nested sampling.
Introduction: the computational problem, Nested sampling: the basic idea, Generating a new object by random sampling,
Monte Carlo sampling of the posterior, How many objects are needed?, Simulated annealing.
Quantification.
Exploring an intrinsically non-uniform prior, Example: ON/OFF switching, Estimating quantities, Final remarks.
A. Gaussian integrals.
A.1 The univariate case.
A.2 The bivariate extension.
A.3 The multivariate generalization.
B. Cox’s derivation of probability.
B.1 Lemma 1: associativity.