Bayesian Reasoning in Data Analysis

The following sections are included:

• Dedication

• Preface

• Contents

The following sections are included:

• 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 frequentistic confidence intervals

• Misunderstandings caused by the standard paradigm of hypothesis tests

• Statistical significance versus probability of hypotheses

The following sections are included:

• 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

• Afraid of ‘prejudices’? Logical necessity versus frequent practical irrelevance of priors

• Recovering standard methods and short-cuts to Bayesian reasoning

• Evaluation of measurement uncertainty: general scheme
  • Direct measurement in the absence of systematic errors
  • Indirect measurements
  • Systematic errors
  • Approximate methods

The following sections are included:

• 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)
  • Variations over a problem to Newton

• Falsificationism and Bayesian statistics

• Probability versus decision

• Probability of hypotheses versus probability of observations

• Choice of the initial probabilities (discrete case)
  • General criteria
  • Insufficient reason and Maximum Entropy

• Solution to some problems
  • AIDS test
  • Gold/silver ring problem
  • Regular or double-head coin?
  • Which random generator is responsible for the observed number?

• Some further examples showing the crucial role of background knowledge

The following sections are included:

• Discrete variables

• Continuous variables: probability and probability density function

• Distribution of several random variables

• Propagation of uncertainty

• 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

• Laws of large numbers

The following sections are included:

• Measurement error and measurement uncertainty
  • General form of 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

The following sections are included:

• Normally distributed observables

• Final distribution, prevision and credibility intervals of the true value

• Combination of several measurements - Role of priors
  • Update of estimates in terms of Kalman filter

• Conjugate priors

• Improper priors — never take models literally!

• Predictive distribution

• Measurements close to the edge of the physical region

• 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

• The Gauss derivation of the Gaussian

The following sections are included:

• Binomially distributed observables
  • Observing 0% or 100%
  • Combination of independent measurements
  • Conjugate prior and many data limit

• The Bayes problem

• Predicting relative frequencies – Terms and interpretation of Bernoulli’s theorem

• Poisson distributed observables
  • Observation of zero counts

• Conjugate prior of the Poisson likelihood

• Predicting future counts

• A deeper look to the Poissonian case
  • Dependence on priors — practical examples
  • Combination of results from similar experiments
  • Combination of results: general case
  • Including systematic effects
  • Counting measurements in the presence of background

The following sections are included:

• Maximum likelihood and least squares as particular cases of Bayesian inference

• Linear fit

• Linear fit with errors on both axes

• More complex cases

• Systematic errors and ‘integrated likelihood’

• Linearization of the effects of influence quantities and approximate formulae

• BIPM and ISO recommendations

• Evaluation of type B uncertainties

• Examples of type B uncertainties

• Comments on the use of type B uncertainties

• Caveat concerning the blind use of approximate methods

• Propagation of uncertainty

• Covariance matrix of experimental results – more details
  • Building the covariance matrix of experimental data
    • Offset uncertainty
    • Normalization uncertainty
    • General case

• Use and misuse of the covariance matrix to fit correlated data
  • Best estimate of the true value from two correlated values
  • Offset uncertainty
  • Normalization uncertainty
  • Peelle’s Pertinent Puzzle

The following sections are included:

• Problem and typical solutions

• Bayes’ theorem stated in terms of causes and effects

• Unfolding an experimental distribution

The following sections are included:

• 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?

• Frequentists and Bayesian ‘sects’
  • Bayesian versus frequentistic methods
  • Subjective or objective Bayesian theory?
  • Bayes’ theorem is not everything

• Biased Bayesian estimators and Monte Carlo checks of Bayesian procedures

• Frequentistic coverage
  • Orthodox teacher versus sharp student - a dialogue by George Gabor

• Why do frequentistic hypothesis tests’often work’?

• Comparing ‘complex’ hypotheses - automatic Ockham’ Razor

• Bayesian networks
  • Networks of beliefs - conceptual and practical applications
  • The gold/silver ring problem in terms of Bayesian networks

The following sections are included:

• Use and misuse of the standard combination rule

• ‘Apparently incompatible’ experimental results

• Sceptical combination of experimental results
  • Application to ɛ′/ɛ
  • Posterior evaluation of σ_i

The following sections are included:

• Usual combination of ‘statistic and systematic errors’

• Sources of asymmetric uncertainties in standard statistical procedures
  • Asymmetric χ² and ‘Δχ² = 1 rule’
  • Systematic effects
    • Asymmetric beliefs on systematic effects
    • Nonlinear propagation of uncertainties

• General solution of the problem

• Approximate solution
  • Linear expansion around E(X)
  • Small deviations from linearity

• Numerical examples

• The non-monotonic case

The following sections are included:

• Frontier physics measurements at the limit to the detector sensitivity

• Desiderata for an optimal report of search results

• Master example: Inferring the intensity of a Poisson process in the presence of background

• Modelling the inferential process

• Choice of priors
  • Uniform prior
  • Jeffreys’ prior
  • Role of priors
  • Priors reflecting the positive attitude of researchers

• Prior-free presentation of the experimental evidence

• Some examples of -function based on real data

• Sensitivity bound versus probabilistic bound

• Open versus closed likelihood

The following sections are included:

• About subjective probability and Bayesian inference

• Conservative or realistic uncertainty evaluation?

• Assessment of uncertainty is not a mathematical game

• Bibliographic note

The following sections are included:

• Bibliography

• Index