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Contents
1 Recipes for a Good Statistical Analysis
2 A Bit of Theory
2. 1 Axiom 1:Probabilities Are in the Range Zero to One
2. 2 Axiom 2:When a Probability Is Either Zero or One
2. 3 Axiom 3:The Sum, or Marginalization, Axiom
2. 4 Product Rule
2. 5 Bayes Theorem
2. 6 Error Propagation
2. 7 Bringing It All Home
2. 8 Profiling Is Not Marginalization
2. 9 Exercises
References
3 A Bit of Numerical Computation
3. 1 Some Technicalities
3. 2 How to Sample from a Generic Function
References
4 Single Parameter Models
4. 1 Step-by-Step Guide for Building a Basic Model
4. 1. 1 A Little Bit of (Science) Background
4. 1. 2 Bayesian Model Specification
4. 1. 3 Obtaining the Posterior Distribution
4. 1. 4 Bayesian Point and Interval Estimation
4. 1. 5 Checking Chain Convergence
4. 1. 6 Model Checking and Sensitivity Analysis
4. 1. 7 Comparison with Older Analyses
4. 2 Other Useful Distributions with One Parameter
4. 2. 1 Measuring a Rate:Poisson
4. 2. 2 Combining Two or More (Poisson) Measurements
4. 2. 3 Measuring a Fraction:Binomial
4. 3 Exercises
References
5 The Prior
5.1 Conclusions Depend on the Prior …
5.1.1 … Sometimes a Lot: The Malmquist-Eddington Bias
5.1.2 … by Lower Amounts with Increasing Data Quality
5.1.3 … but Eventually Becomes Negligible
5.1.4 … and the Precise Shape of the Prior Often Does Not Matter
5. 2 Where to Find Priors
5. 3 Why There Are So Many Uniform Priors in this Book?
5. 4 Other Examples on the Influence of Priors on Conclusions
5. 4. 1 The Important Role of the Prior in the Determination of the Mass
of the Most Distant Known Galaxy Cluster
5. 4. 2 The Importance of Population Gradients for Photometric
Redshifts
5. 5 Exercises
References
6 Multi-parameters Models
6. 1 Common Simple Problems
6. 1. 1 Location and Spread
6. 1. 2 The Source Intensity in the Presence of a Background
6. 1. 3 Estimating a Fraction in the Presence of a Background
6. 1. 4 Spectral Slope:Hardness Ratio
6. 1. 5 Spectral Shape
6. 2 Mixtures
6. 2. 1 Modeling a Bimodal Distribution:The Case of Globular Cluster
Metallicity
6. 2. 2 Average of Incompatible Measurements
6. 3 Advanced Analysis
6. 3. 1 Source Intensity with Over-Poisson Background Fluctuations
6. 3. 2 The Cosmological Mass Fraction Derived from the Cluster’s
Baryon Fraction
6. 3. 3 Light Concentration in the Presence of a Background
6. 3. 4 A Complex Background Modeling for Geo-Neutrinos
6. 3. 5 Upper Limits from Counting Experiments
6. 4 Exercises
References
7 Non-random Data Collection
7. 1 The General Case
7. 2 Sharp Selection on the Value
7. 3 Sharp Selection on the Value, Mixture of Gaussians:Measuring the
Gravitational Redshift
7. 4 Sharp Selection on the True Value
7. 5 Probabilistic Selection on the True Value
7. 6 Sharp Selection on the Observed Value, Mixture of Gaussians
7. 7 Numerical Implementation of the Models
7. 7. 1 Sharp Selection on the Value
7. 7. 2 Sharp Selection on the True Value
7. 7. 3 Probabilistic Selection on the True Value
7. 7. 4 Sharp Selection on the Observed Value, Mixture of Gaussians
7. 8 Final Remarks
Reference
8 Fitting Regression Models
8. 1 Clearing Up Some Misconceptions
8. 1. 1 Pay Attention to Selection Effects
8. 1. 2 Avoid Fishing Expeditions
8. 1. 3 Do Not Confuse Prediction with Parameter Estimation
8. 2 Non-linear Fit with No Error on Predictor and No Spread:
Efficiency and Completeness
8. 3 Fit with Spread and No Errors on Predictor:Varying Physical
Constants?
8. 4 Fit with Errors and Spread:The Magorrian Relation
8. 5 Fit with More Than One Predictor and a Complex Link:Star
Formation Quenching
8. 6 Fit with Upper and Lower Limits:The Optical-to-X Flux Ratio
8. 7 Fit with An Important Data Structure:The Mass-Richness Scaling
8. 8 Fit with a Non-ignorable Data Collection
8. 9 Fit Without Anxiety About Non-random Data Collection
8. 10 Prediction
8. 11 A Meta-Analysis:Combined Fit of Regressions with Different
Intrinsic Scatter
8. 12 Advanced Analysis
8. 12. 1 Cosmological Parameters from SNIa
8. 12. 2 The Enrichment History of the ICM
8. 12. 3 The Enrichment History After Binning by Redshift
8. 12. 4 With An Over-Poissons Spread
8. 13 Exercises
References
9 Model Checking and Sensitivity Analysis
9. 1 Sensitivity Analysis
9. 1. 1 Check Alternative Prior Distributions
9. 1. 2 Check Alternative Link Functions
9. 1. 3 Check Alternative Distributional Assumptions
9. 1. 4 Prior Sensitivity Summary
9. 2 Model Checking
9. 2. 1 Overview
9. 2. 2 Start Simple:Visual Inspection of Real and Simulated Data and
of Their Summaries
9. 2. 3 A Deeper Exploration:Using Measures of Discrepancy
9. 2. 4 Another Deep Exploration
9. 3 Summary
References
10 Bayesian vs Simple Methods
10. 1 Conceptual Differences
10. 2 Maximum Likelihood
10. 2. 1 Average vs.Maximum Likelihood
10. 2. 2 Small Samples
10. 3 Robust Estimates of Location and Scale
10. 3. 1 Bayes Has a Lower Bias
10. 3. 2 Bayes Is Fairer and Has Less Noisy Errors
10. 4 Comparison of Fitting Methods
10. 4. 1 Fitting Methods Generalities
10. 4. 2 Regressions Without Intrinsic Scatter
10. 4. 3 One More Comparison, with Different Data Structures
10. 5 Summary and Experience of a Former Non-Bayesian Astronomer
References
A Probability Distributions
A.1 Discrete Distributions
A.1.1 Bernoulli
A.1.2 Binomial
A.1.3 Poisson
A.2 Continuous Distributions
A.2.1 Gaussian or Normal
A.2.2 Beta
A.2.3 Exponential
A.2.4 Gamma and Schechter
A.2.5 Lognormal
A.2.6 Pareto or Power Law
A.2.7 Central Student-t
A.2.8 Uniform
A.2.9 Weibull
B The Third Axiom of Probability, Conditional Probability,
Independence and Conditional Independence
B.1 The Third Axiom of Probability
B.2 Conditional Probability
B.3 Independence and Conditional Independence
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