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Bayesian Reasoning in High
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Bayesian Reasoning in High
Contents
Subjective probability in Physics? Scientific reasoning in conditions of uncertainty
Uncertainty in physics and the usual methods of handling it
Uncertainty in physics
True value, error and uncertainty
Sources of measurement uncertainty
Usual handling of measurement uncertainties
Probability of observables versus probability of true values
Probability of the causes
Unsuitability of confidence intervals
Misunderstandings caused by the standard paradigm of hypothesis tests
Statistical significance versus probability of hypotheses
A probabilistic theory of measurement uncertainty
Where to restart from?
Concepts of probability
Subjective probability
Learning from observations: the `problem of induction'
Beyond Popper's falsification scheme
From the probability of the effects to the probability of the causes
Bayes' theorem for uncertain quantities: derivation from a physicist's point of view
Afraid of `prejudices'? Inevitability of principle and frequent practical irrelevance of the priors
Recovering standard methods and short-cuts to Bayesian reasoning
Evaluation of uncertainty: general scheme
Direct measurement in the absence of systematic errors
Indirect measurements
Systematic errors
Approximate methods
``Bayesian primer'' - slightly reviewed version of the 1995 DESY/Rome report -
Subjective probability and Bayes' theorem
Orig inal abstract of the ``primer''
Introduction to the ``primer''
Probability
What is probability?
Subjective definition of probability
Rules of probability
Subjective probability and ``objective'' description of the physical world
Conditional probability and Bayes' theorem
Dependence of the probability on the state of information
Conditional probability
Bayes' theorem
Conventional use of Bayes' theorem
Bayesian statistics: learning by experience
Hypothesis test (discrete case)
Choice of the initial probabilities (discrete case)
General criteria
Insufficient reason and maximum entropy
Distributions (a concise reminder)
Random variables
Discrete variables
Continuous variables: probability and density function
Distribution of several random variables
Central limit theorem
Terms and role
Distribution of a sample average
Normal approximation of the binomial and of the Poisson distribution
Normal distribution of measurement errors
Caution
Bayesian inference applied to measurements
Measurement errors and measurement uncertainty
Statistical inference
Bayesian inference
Bayesian inference and maximum likelihood
The dog, the hunter and the biased Bayesian estimators
Choice of the initial probability density function
Difference with respect to the discrete case
Bertrand paradox and angels' sex
Normally distributed observables
Final distribution, prevision and credibility intervals of the true value
Combination of several measurements
Measurements close to the edge of the physical region
Counting experiments
Binomially distributed observables
Poisson distributed quantities
Uncertainty due to systematic errors of unknown size
Example: uncertainty of the instrument scale offset
Correction for known systematic errors
Measuring two quantities with the same instrument having an uncertainty of the scale offset
Indirect calibration
Counting measurements in the presence of background
Bypassing the Bayes' theorem for routine applications
Approximate methods
Linearization
BIPM and ISO recommendations
Evaluation of type B uncertainties
Examples of type B uncertainties
Caveat concerning the blind use of approximate methods
Indirect measurements
Covariance matrix of experimental results
Building the covariance matrix of experimental data
Use and misuse of the covariance matrix to fit correlated data
Bayesian unfolding
Problem and typical solutions
Bayes' theorem stated in terms of causes and effects
Unfolding an experimental distribution
Other comments, examples and applications
Appendix on probability and inference
Unifying role of subjective approach
Frequentists and combinatorial evaluation of probability
Interpretation of conditional probability
Are the beliefs in contradiction to the perceived objectivity of physics?
Biased Bayesian estimators and Monte Carlo checks of Bayesian procedures
Frequentistic coverage
Bayesian networks
Why do frequentistic hypothesis tests `often work'?
Frequentists and Bayesian `sects'
Bayesian versus frequentistic methods
Orthodox teacher versus sharp student - a dialogue by Gabor
Subjective or objective Bayesian theory?
Bayes' theorem is not all
Solution to some problems
AIDS test
Gold/silver ring problem
Further HEP applications
Poisson model: dependence on priors, combination of results and systematic effects
Dependence on priors
Combination of results from similar experiments
Combination of results: general case
Including systematic effects
Is there a signal?
Signal and background: a Mathematica example
Unbiased results
Uniform prior and fictitious quantities
Constraining the mass of a hypothetical new particle: analysis strategy on a toy model
The rules of the game
Analysis of experiment
Naïve procedure
Correct procedure
Interpretation of the results
Outside the sensitivity region
Including other experiments
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Conclusions
About subjective probability and Bayesian inference
Conservative or realistic uncertainty evaluation?
Assessment of uncertainty is not a mathematical game
Bibliography
Subsections
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Introduction
Giulio D'Agostini 2003-05-15