Principles and Basic Applications

**G. D'Agostini
Università ``La Sapienza'' and INFN, Roma, Italia
**

This report introduces general ideas and some
basic methods of the Bayesian probability theory
applied to physics measurements.
Our aim is to make the reader familiar, through examples
rather than rigorous formalism, with concepts such as:
model comparison
(including the automatic *Ockham's Razor* filter provided
by the Bayesian approach); parametric inference;
quantification of the uncertainty
about the value of physical quantities, also taking
into account systematic effects; role of marginalization;
posterior characterization;
predictive distributions;
hierarchical modelling and hyperparameters;
Gaussian approximation of the posterior
and recovery of conventional methods, especially maximum likelihood
and chi-square fits under well defined conditions;
conjugate priors, transformation invariance and maximum
entropy motivated priors; Monte Carlo estimates of expectation,
including a short introduction to Markov Chain Monte Carlo
methods. ^{1}

**Note**: This is an html version of the preprint
of a review article
**published in
Reports on Progress in Physics
66
(2003) 1383**.
See here for printable versions of
the preprint as well as for related publications by the
author.

- Introduction
- Uncertainty and probability
- Rules of probability
- Probability of simple propositions
- Probability of complete classes
- Probability rules for uncertain variables

- Bayesian inference for simple problems

- Inferring numerical values of physics quantities -- General ideas
and basic examples
- Bayesian inference on uncertain variables and posterior characterization
- Gaussian model
- Binomial model
- Poisson model
- Inference from a data set and sequential use of Bayes formula
- Multidimensional case -- Inferring and of a Gaussian
- Predictive distributions
- Hierarchical modelling and hyperparameters
- From Bayesian inference to maximum-likelihood and minimum chi-square model fitting
- Gaussian approximation of the posterior distribution

- Uncertainties from systematic effects
- Reweighting of conditional inferences
- Joint inference and marginalization of nuisance parameters
- Correlation in results caused by systematic errors
- Approximate methods and standard propagation applied to systematic errors

- Comparison of models of different complexity
- Choice of priors - a closer look
- Logical and practical role of priors
- Purely subjective assessment of prior probabilities
- Conjugate priors
- General principle based priors

- Computational issues

- Conclusions
- References
- About this document ...

Giulio D'Agostini 2003-05-13